AP Physics C: Electricity and Magnetism FRQ Room

Analyzing Transient Response in RLC Circuits Using Calculus-Based Modeling

Design an experiment to study transient oscillations in an RLC circuit. Use calculus to model the di

Charge Conservation and Electrostatic Interactions

Charge conservation and electrostatic interactions are fundamental concepts in electromagnetism. Dis

Charge Induction on a Conductor

When a positively charged rod is brought near a neutral conductor, charge induction occurs. Analyze

Charging and Discharging of a Capacitor: Calculus Approach

In this question, you will derive and solve the differential equation governing the charging/dischar

Comprehensive Analysis of Electric Field Gradients in a Complex System

An advanced sensor array is used to map the electric field gradient in a system with multiple charge

Designing an Experiment to Measure the Electric Field due to a Cylindrical Charge Distribution

Design an experiment to measure the electric field around a long, uniformly charged cylinder using a

Determining Electric Potential from Experimental Measurements

A set-up is used to measure the electric potential at various distances from a point charge. The mea

Electric Dipole in a Uniform Electric Field

An electric dipole with dipole moment $$p$$ is placed in a uniform electric field $$E$$ such that th

Electric Field from a Cylindrical Charge Distribution

An infinitely long cylindrical rod of radius $$R = 0.05 \; m$$ carries a uniform volume charge densi

Electric Field from a Uniformly Charged Plane

An infinite plane sheet with uniform surface charge density produces a constant electric field. Anal

Electric Field from a Uniformly Charged Rod

A researcher is studying the electric field produced by a thin rod of length $$L$$, uniformly charge

Electric Field from an Extended Linear Charge Distribution

A thin rod of length $$L = 0.50 \; m$$ has a uniform linear charge density $$\lambda = 2.0*10^{-8} \

Electric Field Gradient and Potential Difference

A group of students measures the spatial distribution of an electric field along a straight line usi

Electric Field Mapping and Superposition

This problem focuses on constructing electric field lines for multiple charge configurations and app

Electric Field Mapping in a Non-Uniform Field

A researcher maps the electric field in a region where the potential is not uniform. The potential i

Electric Field of a Charged Cylinder

A long cylindrical rod of radius $$R = 0.05 \; m$$ has a uniform volume charge density of $$\rho = 4

Electric Field of an Infinite Line Charge

This question guides you through the derivation of the electric field due to an infinite line charge

Electric Flux Graph Analysis using Gauss' Law

A laboratory experiment measured the electric flux $$\Phi$$ (in N·m²/C) through closed surfaces for

Electric Potential and Field from a Point Charge

A point charge of 5 μC is located at the origin. (a) Write the expression for the electric potential

Electric Potential Energy of a Continuous Charge Distribution

A researcher seeks to determine the total electric potential energy stored in a continuous charge di

Electric Potential Energy of a Uniformly Charged Rod

Derive an expression for the electric potential energy stored in a uniformly charged rod and discuss

Electrostatic Force in an Equilateral Triangle of Charges

This problem examines the net electrostatic force on a charge located at one vertex of an equilatera

Electrostatic Force Variation with Distance

An experiment is conducted to verify Coulomb's law by measuring the force between two point charges

Experimental Verification of Charge Conservation in Circuit Dynamics

Design an experiment to verify the conservation of electric charge in a closed circuit containing a

Field and Potential from an Infinite Sheet of Charge

An infinite sheet has a uniform surface charge density of $$\sigma = 4.0 \times 10^{-6} \; C/m^2$$.

FRQ 2: Electric Field Measurement around a Uniformly Charged Sphere

A report claims that the electric field measured outside a uniformly charged sphere obeys the relati

FRQ 8: Investigating Insulator Polarization Effects

A recent experiment claims that an insulating material exhibits zero internal electric field due to

FRQ 12: Evaluating Electric Potential Energy Change in a Particle Accelerator

A research article claims that measured changes in the electric potential energy of particles in an

FRQ 20: Analysis of Measurement Errors in Field Strength from Extended Charge Distribution

A research team reports that the calculated electric field from an extended planar charge distributi

Gauss' Law and Cylindrical Charge Distributions

Consider an infinite line of charge with linear charge density $$\lambda$$ and a cylindrical shell w

Gauss' Law in a Cylindrical Charge Distribution

An infinitely long cylinder of radius R carries a uniform volume charge density ρ. Use Gauss’s law t

Gauss' Law in Practice: Flux Calculation for a Charged Cylinder

Develop an experimental proposal to measure the electric flux through a cylindrical surface surround

Investigating Electric Fields in Conductors vs. Insulators

Design an experiment to compare the electric field distribution around a metallic (conductor) sphere

Investigating the Distance Dependence of Electrostatic Force

A student designs an experiment to measure the electrostatic force between two small charged spheres

Mapping Electric Field in a Complex Geometry via Monte Carlo Integration

An experiment is performed to map the electric field in a region with a complex charge distribution

Mapping Equipotential Lines and Electric Fields Around Complex Charge Distributions

Design an experiment to map equipotential lines around a complex electrode configuration. Use your e

Measuring Electric Field Gradient in a Non-uniform Field

In an experiment designed to determine the gradient of a non-uniform electric field, the experimente

Measuring the Work Done by an Electric Field

In this experiment, a small test charge is released between two parallel plates, where the electric

Potential from a Line of Charge with Variable Density

A rod extends along the x-axis from $$x = 0$$ to $$x = 0.8 \; m$$. Its linear charge density is give

Superposition of Electric Fields in a Composite System

A composite electrical system consists of a point charge and an extended charge distribution. A poin

Transient Analysis in a Discharging RC Circuit

Consider the transient behavior of a discharging capacitor in an RC circuit.

Uniformly Charged Rod Field Calculation

A thin rod of length $$L = 1.00 \; m$$ lies along the x-axis from $$x = 0$$ to $$x = 1.00 \; m$$ and

Work Done by a Uniform Electric Field

A particle with charge $$q = +1.602*10^{-19} \; C$$ is moved over a distance of $$d = 0.10 \; m$$ in

Work Done by Electric Fields in a Particle Accelerator

In designing a particle accelerator, a researcher wants to determine the work done by a constant ele

Work Done in a Non-Uniform Electric Field

This question requires you to compute the work done on a charge moving through a spatially varying (

Analysis of Capacitors in Series and Parallel

A circuit contains three capacitors arranged such that capacitors $$C_1$$ and $$C_2$$ are connected

Analyzing Charge Distribution on a Composite Conductor

A researcher explores a composite conductor comprising regions with different curvatures and materia

Calculus Analysis of Electric Field and Potential in a Non-Uniform Capacitor

A parallel plate capacitor contains a non-uniform dielectric where the permittivity varies with dist

Calculus-Based Analysis of Induced Charge on Shielding Conductors

A point charge is brought near a large, grounded, conductive plate. This induces a non-uniform surfa

Capacitor as a Pressure Sensor

A parallel plate capacitor is used as a pressure sensor. When pressure is applied, one of the plates

Capacitor Charging and Discharging in an RC Circuit

In a series RC circuit, a capacitor with capacitance \(C\) is charged by a constant voltage source \

Capacitor Charging in an RC Circuit with Constant Resistance

A capacitor with capacitance $$C$$ is connected in series with a resistor of resistance $$R$$ and a

Capacitor Discharge Curve and Time Constant Computation

A capacitor in an RC circuit discharges over time, and its voltage follows the equation $$V(t)=V_0 e

Capacitor Energy Derivation by Integration

Derive the expression for the energy stored in a capacitor by integrating the incremental work done

Charge Distribution on an Irregular Conductor

The distribution of charge on an irregularly shaped conductor is non-uniform. Design an experiment t

Derivation of Capacitance Change Due to Temperature-Dependent Dielectric

The dielectric constant of a capacitor material is temperature-dependent, given by $$k(T)=k_i+\delta

Design and Analysis of a Tunable Capacitor Circuit for Frequency Modulation

A tunable capacitor is integrated into an LC oscillator circuit to modulate the produced frequency.

Designing an AC Circuit with a Capacitor and Calculus Analysis

In an AC circuit, the voltage across a capacitor is given by \(V(t) = V_0 * \sin(2 * \pi * f * t)\),

Dielectric Strength and Material Breakdown Analysis

A capacitor with a dielectric material of thickness \(d\) is subject to an electric field. The diele

Discharge Analysis of a Capacitor

A capacitor of capacitance $$C$$, initially charged to a voltage $$V_0$$, is allowed to discharge th

Effect of a Dielectric on Breakdown Voltage

A parallel plate capacitor uses air as the dielectric. The dielectric strength of air is approximate

Electric Field Distribution on a Conducting Surface

A conductor possesses regions with different curvatures including a flat area and a sharply curved t

Electric Field Distribution on a Curved Conductor

Consider a conducting sphere with radius $$R$$ that carries an excess charge $$Q$$. Using Gauss's la

Electric Shielding and Faraday Cage Effectiveness

A research laboratory is designing a Faraday cage to protect sensitive equipment from external elect

Electrostatic Analysis of an Irregular Conductor

An irregularly shaped conductor carries a net charge $$Q$$. Due to its geometry, some regions have h

Electrostatic Induction on a Spherical Conductor

A spherical conductor of radius \(R\) is placed in the vicinity of an external point charge \(Q\) lo

Electrostatic Pressure on a Conductor

When a conductor is charged, an electrostatic pressure is developed on its surface. Answer the follo

Energy Analysis in Charging and Discharging

In an RC circuit, a capacitor is first charged to a voltage $$V$$ and then discharged through a resi

Energy Analysis of a Dielectric-Loaded Capacitor

Consider a parallel plate capacitor with plate area $$A$$, separation $$d$$, and filled completely w

Energy Density Analysis in a Capacitor With a Non-Uniform Electric Field

A capacitor has an electric field that varies with position according to $$E(x) = \frac{V}{d + a \cd

Energy Storage in a Parallel Plate Capacitor

A parallel plate capacitor is used to store energy, and you are tasked with experimentally verifying

Evaluation of Faraday Cage Performance under High-Frequency EMI

A Faraday cage’s performance is tested against high-frequency electromagnetic interference. Using th

Experimental Determination of Dielectric Constant Using a Variable Capacitor

Design an experiment to determine the dielectric constant \(k\) of an unknown material using a varia

Faraday Cage Effectiveness Evaluation

A researcher investigates the effectiveness of a Faraday cage by comparing the external and internal

FRQ 5: Faraday Cage Performance Analysis

A Faraday cage is constructed to protect sensitive electronic equipment from electromagnetic pulses

FRQ 7: Combined Effects of Conductors and Dielectrics

An advanced experiment involves a spherical conductor which is partially filled with a dielectric ma

FRQ 9: Dynamic Change in Capacitance with Variable Plate Area Experiment

A researcher conducted an experiment in which the effective area of one plate in a parallel-plate ca

FRQ 11: Analysis of a Multi-Layer Dielectric Capacitor

A parallel plate capacitor is constructed with two layers of dielectric materials. The first layer h

FRQ 13: Theoretical Analysis of Electric Field Shielding in Layered Conductors

Multiple concentric conductive shells are used to shield a sensitive instrument from external electr

FRQ 15: Dielectric Strength Analysis

In a materials laboratory, dielectric breakdown measurements are performed on various insulating mat

FRQ 17: Optimization of Capacitor Design for Maximum Energy Density

A company is designing a parallel plate capacitor for high-energy storage applications. The energy s

FRQ 18: Non-linear Capacitance in Varactor Diodes

In a semiconductor laboratory, researchers study varactor diodes where the capacitance is non-linear

Geometric Effects on Capacitance in a Non-Parallel Plate Configuration

Two conductive plates form a capacitor but are not perfectly parallel. They form an angle \(\theta\)

Impact of Conductor Geometry on Shielding Efficiency

A metal enclosure is used for electric shielding but has small apertures in its structure. Evaluate

Impact of Electrode Geometry on Capacitance

An experiment compares the capacitance of a capacitor with differing electrode geometries. The measu

Investigating the Role of Curvature on Surface Charge Density in Conductors

A researcher studies a set of conductors with different curvatures (from flat plates to spheres) and

Measuring the Electric Field of a Conducting Sphere

A researcher is measuring the electric field near the surface of a uniformly charged spherical condu

Modeling Electric Field Interactions Between Two Conductors

When two conductors are placed in close proximity, their electric fields interact. Design an experim

Modeling Surface Charge Density and Curvature

For a conductor with non-uniform curvature, the local surface charge density $$\sigma(x)$$ is expect

Modelling Electric Fields Around a Spherical Conductor

A spherical conductor carrying a net charge is analyzed to determine the electric field both inside

Non-Ideal Capacitor Discharge in a Resistive-Capacitive Network

A non-ideal capacitor discharges through a resistor and also loses charge via a constant leakage cur

Optimization of Capacitor Design with Dielectric Materials

A researcher is tasked with optimizing the design of a parallel plate capacitor to maximize capacita

Oscillatory Behavior in LC Circuits

An LC circuit can exhibit oscillatory behavior, and you are to design an experiment to measure the o

Parallel Plate Capacitor: Effect of Dielectric Insertion

A parallel plate capacitor initially charged to a voltage $$V$$ is modified by the slow insertion of

RC Circuit: Charging and Discharging Analysis with Differential Equations

In an RC circuit, a capacitor charges through a resistor from a DC source. Develop and solve the dif

Surface Charge Density on a Non-Uniform Conductor

A conductor with regions of differing curvature will have a non-uniform surface charge density $$\si

Transient Analysis of Capacitor Charging in RC Circuits

Transient analysis of capacitor charging provides insight into the dynamic behavior of RC circuits.

Transient Response of Electric Field in Shielded Conductors Under Pulsed Voltage

A pulsed voltage is applied to a shielded conductor. The transient response of the internal electric

Tuning Characteristics of a Variable Capacitor in Radio Circuits

Variable capacitors are used in LC circuits for radio tuning. The resonant frequency is given by $$f

Variable Resistor in an RC Charging Circuit

Consider an RC circuit in which the resistor is not constant but increases linearly with time accord

AC Circuit: Impedance and Power Factor Analysis

An AC circuit operates at a frequency of $$f = 60$$ Hz and consists of a resistor (R = $$10$$ $$\Ome

Analysis of a Non-Ideal Battery's Internal Resistance

A battery’s terminal voltage was measured under various load currents to study the effect of interna

Analysis of AC Circuit Frequency Effects

In an AC circuit, the voltage is described by $$V(t) = V_0 \sin(2\pi f t)$$ with $$V_0 = 10\,V$$ and

Analysis of Resonance in an RLC Circuit

Consider a series RLC circuit with R = 10 Ω, L = 0.5 H, and C = 20 μF. Analyze the circuit’s behavio

Capacitor Discharge Analysis

A capacitor initially charged to 12 V discharges through a resistor R = 50 Ω. Given that the capacit

Capacitors in Series and Parallel Combination in Sensor Circuits

In a sensor circuit, four capacitors are arranged such that three capacitors (C1, C2, C3) are connec

Capacitors in Series: Voltage Division Analysis

A series circuit contains three capacitors with values $$C_{1}=2\,\mu F$$, $$C_{2}=3\,\mu F$$, and $

Comparative Analysis of AC and DC Signals using a Function Generator

A function generator is used to produce both DC and AC signals. Compare the two types of signals in

Complex Multi-Loop Circuit Analysis

Consider a circuit with two loops sharing a common resistor. Loop 1 contains a voltage source \(V_1

Design and Analysis of an RC High-Pass Filter

A researcher designs an RC high-pass filter using a resistor of $$R=2000\,\Omega$$ and a capacitor o

Design of a Low-Pass RC Filter

Design a low-pass RC filter to attenuate frequencies above 1 kHz. (a) Starting from the voltage divi

Designing an RC Oscillator Circuit

Design an RC oscillator circuit that generates a sine wave output. (a) Derive the expression for the

Determining the EMF and Internal Resistance of a Battery

This question explores the relationship between a battery's EMF, its terminal voltage under load, an

Electromechanical System Experiment with Erroneous Energy Conversion Assumption

A researcher builds an electromechanical system in which a motor converts electrical energy into mec

Energy Efficiency Analysis of Power Supplies

An engineer claims that the energy efficiency of a power supply can be determined by comparing the t

Energy Storage in Capacitors and Inductors

Using calculus, derive the expressions for the energy stored in (a) a capacitor with capacitance C c

Evaluating Power Factor in an AC Circuit

Measurements were taken in an AC circuit to determine its power factor. The table below includes pha

Frequency Dependence of Capacitive Reactance

Investigate how the capacitive reactance in an AC circuit depends on frequency, and use calculus to

FRQ 8: Kirchhoff's Current Law at a Node

At a junction in a circuit, three branches converge. Two of the branches have measured currents of $

FRQ 12: Impedance Calculation in an AC Circuit

Consider an AC circuit with a resistor of $$50\,\Omega$$ in series with an inductor of $$0.2\,H$$ op

FRQ 16: Circuit Measurement Techniques

In a laboratory setting, students are asked to measure voltage, current, and resistance in a circuit

Graphical Analysis of Capacitor Charging Data

A set of experimental data for a capacitor charging in an RC circuit is provided via a graph. Analyz

Ideal Versus Non-Ideal Circuit Component Analysis

Theoretical models using ideal circuit components often predict behavior that deviates from experime

Investigating the Effect of Variable Resistance on Circuit Current

A variable resistor (potentiometer) is used in a DC circuit powered by a 12 V battery. Experimental

Kirchhoff’s Current Law in a Multi-Branch Circuit

In an experimental setup, measurements at a node show two currents entering as \(I_1 = 2.5\,A\) and

Kirchhoff's Current Law Node Analysis

A node in a circuit receives current from a 20 V source and splits into three branches with resistor

Kirchhoff's Voltage Law in a Two-Loop Circuit with Time-Dependent Battery Resistance

A researcher analyzes a two-loop circuit where two loops share a resistor. The circuit contains four

Misinterpretation of Impedance in High-Frequency Circuits

A researcher performs an experiment on a high-frequency AC circuit designed to emulate a transmissio

Non-Ideal Battery EMF Measurement with Internal Resistance Error

A researcher attempts to measure the electromotive force (EMF) of a battery by connecting a voltmete

Parallel Circuit Analysis Using Kirchhoff's Current Law

A DC circuit consists of three resistors connected in parallel to a 24 V battery. The resistor value

Parallel Circuit Analysis with Variable Resistor

Consider a circuit with three resistors connected in parallel to an 18 V battery. The resistors have

Power Dissipation Analysis in a Resistor

In a resistor with \(R = 8\,\Omega\), the current is time-dependent and given by the function $$I(t)

RC Charging Circuit Analysis

In an RC circuit, a capacitor with capacitance $$C=10\,\mu F$$ is charged through a resistor with $$

Resistance Measurements and Conductor Geometry

A uniform wire with a length L = 2 m and cross-sectional area A = 1 mm² (1×10⁻⁶ m²) is made of a mat

RLC Circuit Resonance

An RLC series circuit exhibits resonance at a specific frequency. Analyze the resonance behavior in

Series Circuit Analysis

A simple series circuit consists of a battery with an electromotive force (EMF) of 12 V and an inter

Series Circuit Analysis with Battery Internal Resistance

A student sets up an experiment in which a battery with an open-circuit electromotive force (EMF) V₀

Series Circuit Analysis with Time-Varying Source

Analyze a resistive series circuit with three resistors and a battery whose voltage is modulated by

Series Circuit with Non-Ideal Battery Analysis

Consider a circuit consisting of a non-ideal battery with electromotive force $$\mathcal{E}$$ and in

Stability Analysis of an RC Oscillator

An RC oscillator generates oscillations with a frequency given by $$f = \frac{1}{2\pi R C}$$. Due to

Temperature Effects on Resistance

A researcher investigates how temperature affects the resistance of a thermistor. The experiment mea

Time-Averaged Power in an AC Circuit

An AC voltage source produces a voltage given by $$V(t)=20\times\cos(100\pi\,t)$$ (in volts) and is

Transient Analysis in an RL Circuit

A circuit comprising a resistor and an inductor (an RL circuit) is subject to a DC voltage step. Ana

Transient Response in a Non-Ideal RLC Filter

A series RLC circuit is used as a filter in which transient responses are critical. Analyze the circ

Transient Response in an RL Circuit

Consider an RL circuit where an inductor $$L = 0.5\,H$$ and a resistor $$R = 20\,\Omega$$ are connec

Transient Response in an RL Circuit with Neglected Inductive Kickback Effects

A student investigates the transient response of an RL circuit by abruptly opening a switch and obse

Transient Response of an RC Circuit

An RC circuit with a resistor \(R = 10\,\Omega\) and a capacitor \(C = 2\times10^{-3}\,F\) is connec

Voltage Divider with Variable Resistor

A voltage divider circuit is built using a fixed resistor and a variable resistor. Investigate how t

Analysis of Forced Oscillations in a Magnetic Pendulum

A novel experiment involves a magnetic pendulum oscillating within a time-varying external magnetic

Analysis of Torque on a Wire in a Magnetic Field

A wire carrying current is placed in a magnetic field and experiences a torque. A graph of applied t

Analyzing a Magnetic Resonance Experiment

In a magnetic resonance experiment, a sample is placed in a uniform static magnetic field of 1.5 T a

Application of Ampère’s Law in a Toroid

A toroid with a rectangular cross-section has 500 turns and a mean radius of 0.10 m and carries a cu

Application of the Biot–Savart Law for a Finite Wire

Consider an experiment to measure the magnetic field at a point P due to a finite, straight wire of

Application of the Biot–Savart Law to a Finite Wire

A finite straight wire carrying a current produces a magnetic field at a point offset from its midpo

Charged Particle in a Spatially Varying Magnetic Field

A charged particle with mass $$m$$ and charge $$q$$ moves in a magnetic field that varies spatially

Charged Particle in Combined Electric and Magnetic Fields

In a complex experimental setup, a charged particle is exposed simultaneously to a uniform electric

Comparison of Magnetic Fields Inside and Outside a Solenoid

Measurements were taken of the magnetic field in and around a long solenoid. The data in the table b

Current Variation and Magnetic Field Strength

In an experiment, the magnetic field is measured at a fixed distance from a straight conductor while

Design Considerations for an Electric Motor Component

Electric motors use current-carrying loops in magnetic fields to generate torque. Consider the desig

Designing a Magnetic Crane with Controlled Current

A magnetic crane used in industrial lifting uses large current-carrying conductors to generate a str

Designing and Analyzing a Helmholtz Coil Arrangement

In a Helmholtz coil configuration, two identical circular coils of radius $$R$$ are separated by a d

Determining the Magnetic Dipole Moment of a Current Loop

A circular current loop of radius 0.08 m carries a current of 7 A. (a) Derive an expression for the

Determining Torsional Stiffness from Angular Deflection Data

A metal wire is twisted by applying varying torques. The resulting angular deflection (in radians) w

Determining Torsional Stiffness of a Twisted Wire

A wire is twisted by applying a torque, leading to an angular displacement. The relation is given by

Determining Torsional Stiffness Using Magnetic Torque

A current-carrying wire is twisted by an external magnetic field, generating a measurable torque. (

Effect of Loop Area on Solenoid Magnetic Field

In an experimental study, a researcher examines how the area of a loop placed inside a solenoid affe

Evaluating Magnetic Field Distribution in a Halved Solenoid

A long solenoid is physically divided into two equal halves. (a) Using integration, derive express

Evaluating the Applicability of the Biot-Savart Law

A recent paper asserts that the Biot-Savart Law, given by \(\vec{B} = \frac{\mu_0}{4 * \pi} \int \fr

Experimental Analysis of Magnetic Field Variation with Distance

In an experiment, the magnetic field around a long, straight conductor was measured at different dis

Experimental Investigation of Force Between Parallel Current-Carrying Wires

An experiment is conducted to verify the force between two parallel current-carrying wires. The wire

Force Between Two Parallel Current-Carrying Wires

Derive and analyze the magnetic force between two parallel current-carrying wires using both theoret

Force Between Two Parallel Current-Carrying Wires

Two long, parallel wires carry currents I₁ and I₂ and are separated by a distance d. (a) Derive th

Force on a Current-Carrying Wire

Investigate the force experienced by a current-carrying wire in a uniform magnetic field and analyze

Induced EMF and Changing Magnetic Flux in a Rotating Loop

A rectangular loop with an area of $$A = 0.05\,m^2$$ rotates in a uniform magnetic field of 0.1 T at

Induced EMF in a Rotating Loop

A rectangular loop of area $$A$$ rotates with a constant angular velocity $$\omega$$ in a uniform ma

Induced EMF in a Time-Varying Solenoid

A solenoid with 800 turns, a length of 1.0 m, and a cross-sectional area of 0.02 m² carries a time-v

Investigation of Magnetic Field Lines Visualized With Iron Filings

In a demonstration experiment, a bar magnet is placed underneath a sheet of paper and iron filings a

Investigation of Torsional Stiffness Using a Twisted Wire

A wire is subjected to a torsional force generated by magnetic interactions, and its torsional stiff

Lorentz Force on a Particle at an Oblique Angle

A charged particle with charge q enters a uniform magnetic field B with an initial velocity v that m

Magnet and Current Interactions in Particle Detection

In a particle detector, a solenoid produces a uniform magnetic field of 0.30 T. A charged particle w

Magnetic Dipole Moment from Torque Measurements

A current loop experiences a maximum torque when placed in a uniform magnetic field. (a) Derive an

Magnetic Dipole-Dipole Interaction Energy

Consider two magnetic dipoles, each with magnetic moment $$m$$, separated by a distance $$d$$. Using

Magnetic Field at the Center of a Circular Loop

A circular loop of radius $$R$$ carries a steady current $$I$$. (a) Derive the expression for the ma

Magnetic Field Distribution around a Bar Magnet

A bar magnet produces a magnetic field that is strongest near its poles. Explore the variation in fi

Magnetic Field from a Straight Conductor

A long, straight conductor carries a constant current. Using the formula $$B = \frac{\mu_0 * I}{2\pi

Magnetic Field in a Toroid

A toroid is a solenoid bent into a circular shape. Consider a toroid with inner radius $$a$$, outer

Magnetic Field in a Toroid

A toroid is constructed by winding a wire into a doughnut-shaped coil with N turns. The toroid has a

Magnetic Field Inside a Solenoid

A solenoid is a coil of wire with a large number of turns. Consider a solenoid of length $$L$$, with

Magnetic Field Mapping around a Straight Conductor

An experiment was conducted to measure the magnetic field produced by a straight current-carrying co

Magnetic Field Mapping in a Non-Uniform Field

An experiment measures the magnetic field at various distances from a single current-carrying wire.

Magnetic Field of a Bar Magnet

A bar magnet has a north and a south pole and produces a dipolar magnetic field. (a) Describe qual

Magnetic Field of a Finite Wire Segment

Using the Biot–Savart Law, derive the expression for the magnetic field at a point due to a finite-l

Magnetic Field of a Straight Conductor

Consider an infinitely long straight wire carrying a constant current. Use the standard formula deri

Magnetic Field of a Straight Conductor: Right-Hand Rule and Calculation

A long, straight wire carries a current of 3 A in the positive x-direction. (a) Use the right-hand r

Magnetic Field on the Axis of a Circular Current Loop

Using the Biot-Savart Law, derive and analyze the expression for the magnetic field at points along

Magnetic Field within a Solenoid

A solenoid 0.50 m in length contains 1000 turns and carries a current of 2.0 A. Using the formula $$

Optimization in Electromagnet Design

An electromagnet is constructed by winding N turns of wire around a soft iron core. The magnetic fie

Particle Dynamics in Combined Electric and Magnetic Fields

An experiment studied the trajectory of a charged particle moving through a region with both electri

Superimposed Magnetic Fields from Concentric Loops

Two concentric circular loops with radii $$R_1$$ and $$R_2$$ carry currents $$I_1$$ and $$I_2$$ resp

Torque on a Current Loop

A rectangular loop carrying a current is placed in a uniform magnetic field. (a) Derive an express

Torque on a Current Loop in an Electric Motor Design

An engineering report on electric motor efficiency claims that the torque produced by a current loop

Torque on a Current-Carrying Loop

In the development of electric motors, the phenomenon of torque acting on current-carrying loops is

Torque on a Current-Carrying Loop in a Magnetic Field

A rectangular current-carrying loop is placed in a uniform magnetic field. The torque on the loop is

Verification of Ampère's Law in a Laboratory Setting

A laboratory report claims that Ampère's Law \(\oint \vec{B} \cdot d\vec{l} = \mu_0 * I_{enc}\) hold

Analysis of Ampere's Law in a Capacitor Charging Circuit

An experiment is performed where a loop of wire is placed around a charging capacitor to measure the

Analysis of Energy in an Inductor and LR Circuit

An LR circuit, consisting of an inductor with inductance L and a resistor R, is connected to a const

Analysis of Gauss's Law using a Spherical Charge Distribution

This problem applies Gauss's Law to calculate the electric field outside a uniformly charged sphere.

Analysis of Transformer Primary and Secondary Circuits

A transformer has a primary coil with $$N_p$$ turns and a secondary coil with $$N_s$$ turns, with an

Analyzing Back EMF in an LR Circuit

Consider an LR circuit where, after a steady state current has been established, the applied voltage

Analyzing Displacement Current in a Charging Capacitor

In a parallel-plate capacitor being charged by a current $$I$$, the electric field between the plate

Back EMF in an Electric Motor

An electric motor develops a back electromotive force (EMF) as it speeds up. A graph is provided tha

Calculating Coiled Inductance via Magnetic Flux

A solenoid with $$N$$ turns has a cross-sectional area $$A$$ and is placed in a uniform magnetic fie

Critical Review of Transformer Data

An experiment on transformers claims high efficiency in flux transfer between the primary and second

Designing an Electromagnetic Generator

An electromagnetic generator is designed with a rotating coil placed in a uniform magnetic field. Th

Designing LR Circuit Transient Analysis

This problem explores the transient response of an LR circuit when a constant voltage is applied.

Dynamic Analysis of Energy Transfer in an LC Circuit

This problem explores the oscillatory energy exchange in an LC circuit and requires both derivation

Eddy Current Braking Analysis

A rotating metallic disk is subjected to a magnetic field, and eddy currents are induced which produ

Electromagnetic Braking System: Analysis and Design

Electromagnetic braking utilizes Lenz's law where induced currents produce forces that oppose the mo

Electromagnetic Induction from a Non-Uniform Magnetic Field

Consider a long, straight wire carrying a time-dependent current described by I(t) = I₀ * e^(−α*t),

Electromagnetic Oscillator Simulation

A circuit is designed incorporating an LC oscillator with a resistor to simulate damping, resulting

Electromagnetic Waves and Maxwell’s Equations

This problem explores the derivation of the electromagnetic wave equation in vacuum starting from Ma

Energy Storage in an Inductor

An inductor with inductance $$L=0.50\;H$$ carries a time-dependent current given by $$I(t)=5t\;(A)$$

Experimentally Measuring Magnetic Field Variation

A coil is used to measure the magnetic flux as a magnet is moved relative to it. The following table

Faraday's Law and a Sliding Bar on Rails

A metal bar of length $$l$$ slides on a pair of parallel, conducting rails with a constant velocity

Gauss's Law for Magnetic Fields

Gauss's Law for magnetic fields indicates that the net magnetic flux through any closed surface is z

Gauss's Law for Magnetic Fields: A Theoretical Examination

Consider a closed, irregularly shaped surface placed in a uniform magnetic field. Address the follow

Induced EMF in a Rotating Loop

A rectangular loop of wire with area A rotates with a constant angular speed $$\omega$$ in a uniform

Induced EMF in a Time-Varying Magnetic Environment

A parallel-plate capacitor with plate area A is subjected to a time-varying magnetic field given by

Induction Coil Experiment and Lenz’s Law

In an experiment, a rectangular coil is rapidly inserted into a solenoid connected to a galvanometer

Investigating Eddy Currents in a Conducting Tube

A magnet is dropped through a non-magnetic, conducting tube. The induced eddy currents in the tube g

Investigating the Impact of Angle on Magnetic Flux

A flat loop of area $$A$$ is placed in a uniform magnetic field of strength $$B$$. The loop is rotat

Lenz's Law in Magnetic Levitation Systems

Magnetic levitation (maglev) systems use time-varying magnetic fields to induce currents in conducti

LR Circuit Transient Response

Consider an LR circuit comprising an inductor with inductance $$L$$ and a resistor of resistance $$R

Magnetic Field Calculation in a Solenoid

A long solenoid with a total of $$N$$ turns and length $$l$$ carries a steady current $$I$$. The sol

Magnetic Levitation Using Electromagnetic Induction

A magnetic levitation (maglev) system uses electromagnets to levitate a train. As the train moves, a

Maxwell’s Equations – Gauss’s Law

Gauss’s Law is one of Maxwell’s equations and relates the electric flux through a closed surface to

Maxwell's Equations and Electromagnetic Wave Propagation

Electromagnetic waves in free space are described by Maxwell's Equations. Answer the following:

Measuring Magnetic Flux Variations

A researcher rotates a coil in a uniform magnetic field to study how the magnetic flux changes with

Oscillations in a Series LC Circuit

A series LC circuit is built with an inductor of L = 0.5 H and a capacitor of C = 200 μF (200×10⁻⁶ F

Resonant Behavior in an LC Circuit

A researcher sets up an LC circuit with an inductor (L) and a capacitor (C) connected in series to s

Resonant Frequency Determination in an LC Circuit

An LC circuit is constructed using a capacitor of capacitance $$C = 100\,\mu F$$ and an unknown indu

Rotating Coil in a Uniform Magnetic Field

A rectangular coil rotates in a uniform magnetic field. The magnetic flux through the coil is given

Rotating Loop EMF Analysis

Consider a rectangular conducting loop of area $$A$$ rotating in a uniform magnetic field $$B$$ with

Sinusoidal EMF from an Alternating Current Generator

A researcher investigates an AC generator where a coil rotates in a magnetic field, producing a sinu

Sliding Conductor in a Magnetic Field

A researcher investigates electromagnetic induction by moving a straight conductor of length $$l$$ a

Time-Varying Magnetic Fields and Induced Currents in a Solenoid

A long solenoid is subjected to an external magnetic field varying with time as $$B(t) = B_0 \cos(\o

Transformer Voltage Regulation Experiment

An experiment is conducted to evaluate a transformer’s voltage regulation by measuring the input vol

Transient Analysis in an LR Circuit

Consider an LR series circuit with resistance $$R$$ and inductance $$L$$. At time $$t = 0$$, a switc

Transient Response in an LR Circuit

Consider an LR circuit consisting of an inductor $$L$$ and a resistor $$R_{loop}$$ connected in seri

Variable Magnetic Field Experiment

In a solenoid experiment, the magnetic field $$B$$ inside the solenoid varies with time according to