AP Physics C: E&M Practice Questions & Quizzes

Unit 8 · Electric Charges, Fields, and Gauss’s Law

• BeginnerAn uncharged, solid, spherical conducting shell has an inner radius \(R_1\) and an outer radius \(R_2\). A small positive point charge \(+Q\) is place…
• BeginnerTwo point charges, \(+Q\) and \(-Q\), are fixed on the x-axis at \(x = -d\) and \(x = +d\), respectively. The charges can be treated as ideal point ch…
• BeginnerTwo small conducting spheres are separated by a center-to-center distance \(d\). Sphere 1 carries a charge \(q_1\) and sphere 2 carries a charge…
• BeginnerTwo fixed point charges separated by a distance \(r\) in a vacuum exert an electrostatic force of magnitude \(F_0\) on each other. The charges are the…
• BeginnerA student performs two separate charging experiments using identical, initially uncharged electroscopes and identical negatively charged rods. In Expe…
• BeginnerAn infinitely long, straight line of charge carries a uniform positive linear charge density \(\lambda\). A coaxial cylindrical Gaussian surface of ra…
• BeginnerAn electron of mass \(m_e\) and charge magnitude \(e\) is released from rest in a region containing a uniform electric field of magnitude \(E\) direct…
• BeginnerTwo point charges, one with charge \(+q\) and the other with charge \(-q\), are held fixed at a separation distance \(d\) in a vacuum. What is the mag…
• BeginnerA positively charged glass rod is brought near two small, neutral scraps of paper of equal size: Scrap 1 is made of a conducting material, and Scrap 2…
• BeginnerA uniform electric field of magnitude \(E\) is directed parallel to the central axis of symmetry of an open hemispherical surface of radius \(R\), ent…
• IntermediateA student conducts an experiment to determine the unknown positive charge \(q\) on a small sphere, Sphere A. Sphere A has a mass \(m = 1.0 \text{ g}\)…
• IntermediateThree point charges are fixed along the \(x\)-axis: a charge \(-2q\) is located at the origin, and two charges each of charge \(+q\) are located at…
• IntermediateTwo positive point charges, \(q_1 = Q\) and \(q_2 = \beta Q\) (where \(\beta > 1\) is a dimensionless constant), are fixed on the \(x\)-axis at…
• IntermediateA thin circular ring of radius \(R\) carries a total positive charge \(Q\) distributed uniformly along its circumference. Point \(P\) lies on the cent…
• IntermediateA thin, nonconducting rod bent into a semicircle of radius \(R\) carries a total charge \(Q\) distributed uniformly along its length. A point charge \…
• IntermediateA thin, nonconducting rod of length \(R\) carries a total positive charge \(Q\) distributed uniformly along its length. The rod is centered at the ori…
• IntermediateThree point charges are fixed in the xy-plane: a charge \(+2q\) is located at the origin \((0,0)\), a charge \(-q\) is located at \((a,0)\), and a cha…
• IntermediateA point charge \(q\) is fixed on the central symmetry axis of a flat, circular disk of radius \(R\) at a perpendicular distance \(d\) from the center…
• IntermediateTwo small conducting spheres, \(A\) and \(B\), each of mass \(m\), are placed on a frictionless insulating surface separated by a distance \(d\) in a…
• IntermediateA thin, flat disk of radius \(R\) lies in the \(xy\)-plane centered at the origin and carries a uniform surface charge density \(\sigma\). Which of th…
• AdvancedA thick, non-conducting spherical shell has an inner radius \(R\) and an outer radius \(2R\). The shell has a non-uniform volume charge density…
• AdvancedA very long, nonconducting cylindrical shell of inner radius \(R\) and outer radius \(2R\) has a non-uniform volume charge density given by…
• AdvancedA thin, nonconducting rod of length \(L\) lies along the \(x\)-axis with its left end at the origin \(x = 0\) and its right end at \(x = L\). The rod…
• AdvancedA point charge \(q\) is located on the axis of symmetry of an open paraboloid surface, at a distance \(d\) directly below the center of its circular o…
• AdvancedA thin, nonconducting ring of radius \(R\) carries a total positive charge \(Q\) distributed uniformly along its circumference. A small test particle…
• AdvancedAn insulating solid sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = a r^2\) for \(0 \le r \le R\), where \(a\) is…
• AdvancedAn infinite planar slab of thickness \(d\) is aligned parallel to the \(yz\)-plane between \(x = 0\) and \(x = d\). The slab contains a non-uniform vo…
• AdvancedAn infinitely long, solid insulating cylinder of radius \(R\) carries a non-uniform volume charge density given by…
• AdvancedAn infinitely long non-conducting solid cylinder of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0 \dfrac{r}{R}\) fo…
• AdvancedA thin, rigid insulating ring of radius \(R\) lies in the \(xy\)-plane and carries a total positive charge \(Q\) distributed uniformly along its circu…

Quizzes for this unit

• AP Physics C: E&M Concept Mastery Assessment 5 questions Take quiz →

Unit 9 · Electric Potential

• BeginnerAn uncharged, solid, spherical conducting shell has an inner radius \(R_1\) and an outer radius \(R_2\). A small positive point charge \(+Q\) is place…
• BeginnerTwo point charges, each with charge \(+Q\), are fixed on the y-axis at \(y = +a\) and \(y = -a\). Point P is located on the positive x-axis at a dista…
• BeginnerTwo point charges with charge magnitudes \(+q\) and \(+3q\) are fixed a distance \(d\) apart. The electric potential energy of this two-charge system…
• BeginnerAn isolated positive point charge \(+Q\) is fixed in space. Point \(P\) is located a distance \(r\) from the charge, with the electric potential defin…
• BeginnerA proton with charge \(e\) and mass \(m\) is accelerated from rest through a potential difference \(\Delta V\), reaching a final speed \(v_0\). An alp…
• BeginnerIn the injection stage of a mass spectrometer, an ion with charge \(q\) and mass \(m\) is accelerated from rest through a constant electric potential…
• BeginnerA particle carrying charge \(-q\), where \(q > 0\), is displaced a distance \(d\) in the direction opposite to a uniform electric field of magnitud…
• BeginnerA region of space contains a uniform electric field, represented by four parallel equipotential lines perpendicular to the x-axis. The lines are separ…
• BeginnerA positive point charge \(+q\) is located a distance \(r\) from a fixed negative point charge \(-Q\). The zero of electric potential energy is defined…
• BeginnerThree identical point charges, each with charge \(+q\), are initially fixed at the vertices of an equilateral triangle with side length \(d\). The dis…
• IntermediateA thin, nonconducting disk of radius \(R\) is located in the \(xy\)-plane with its center at the origin. The disk has a uniform surface charge density…
• IntermediateA small particle of mass \(m\) and positive charge \(+q\) is launched from a very large distance (assume \(r \to \infty\)) with initial speed \(v_0\)…
• IntermediateA proton of mass \(m_p = m\) and charge \(q_p = +e\) and a deuteron of mass \(m_d = 2m\) and charge \(q_d = +e\) are each accelerated from rest throug…
• IntermediateA small particle with mass \(m\) and positive charge \(q\) is projected from a very large distance directly toward a fixed, stationary point charge al…
• IntermediateTwo point particles with identical positive charges \(q\) are held at rest a distance \(d\) apart in isolated space. Particle 1 has mass \(m\) and Par…
• IntermediateA thin, flat, nonconducting disk of radius \(R\) lies in the \(xy\)-plane with its center at the origin. A total positive charge \(Q\) is distributed…
• IntermediateAn electric field directed along the \(x\)-axis has a constant value \(E_x = +E_0\) in the region \(0 < x < d\) and a constant value…
• IntermediateAn infinitely long, thin cylindrical shell of radius \(R\) carries a uniform positive linear charge density \(\lambda\). Point 1 is located at a radia…
• IntermediateThree thin, concentric, conducting spherical shells have radii \(R\), \(2R\), and \(3R\), and carry net charges \(+q\), \(+2q\), and \(-3q\), respecti…
• IntermediateThe electric potential in a region of space, expressed in two-dimensional polar coordinates \((r, \theta)\), is given by…
• AdvancedA region of space has an electric potential that varies with position in the \(xy\)-plane. A contour map of the equipotential lines is shown in Figure…
• AdvancedIn a classical model of the hydrogen atom, an electron of charge \(-e\) and mass \(m\) orbits a stationary proton of charge \(+e\) in a circular path…
• AdvancedAlong a segment of the \(x\)-axis, a fixed charge distribution creates an electric potential given by \(V(x) = V_0 + b x^2\), where \(V_0\) and \(b\)…
• AdvancedIn a spherically symmetric region of space, the electric potential as a function of radial distance \(r\) from the origin is given by…
• AdvancedA particle of mass \(m\) and positive charge \(+q\) is constrained to move in the \(xy\)-plane within a region of electrostatic potential given by…
• AdvancedA long, nonconducting solid cylinder of radius \(R\) carries a uniform volume charge density \(\rho\). The electric potential at the surface of the cy…
• AdvancedFour point charges are fixed in the \(xy\)-plane on the coordinate axes at a distance \(a\) from the origin, as shown in the figure. Two positive char…
• AdvancedAn uncharged, solid conducting sphere of radius \(R\) contains an off-center spherical cavity of radius \(r_0\). A point charge \(+q\) is fixed inside…
• AdvancedA thin wire carrying a total charge \(Q\) is wound into a flat, tightly packed spiral with inner radius \(a\) and outer radius \(b\). The winding turn…
• AdvancedA solid insulating sphere of radius \(R\) carries a net charge \(Q\) distributed uniformly throughout its volume. The electric field energy density st…

Unit 10 · Conductors and Capacitors

• BeginnerAn uncharged, solid, spherical conducting shell has an inner radius \(R_1\) and an outer radius \(R_2\). A small positive point charge \(+Q\) is place…
• BeginnerA parallel-plate capacitor with plate separation \(d\) is connected to an ideal battery that maintains a constant potential difference \(V_0\). With a…
• BeginnerA solid conducting sphere initially carries a net positive charge \(+Q\) and sits isolated in space. The sphere is then connected to Earth, which acts…
• BeginnerAn ideal parallel-plate capacitor with plate area \(A\) and plate separation \(d\) has initial capacitance \(C_0\). A slab of uniform dielectric mater…
• BeginnerAn uncharged solid conducting sphere is fixed in space near a negative point charge \(-Q\). Which of the following correctly describes the electric po…
• BeginnerAn ideal parallel-plate capacitor consists of two large, oppositely charged parallel conducting plates separated by a distance \(d\). In the region be…
• BeginnerAn air-filled parallel-plate capacitor with plate separation \(d\) is connected to a battery that maintains a constant potential difference \(V_0\). W…
• BeginnerTwo identical, isolated conducting spheres, A and B, carry initial charges of \(+4Q\) and \(-2Q\), respectively. The spheres are brought into brief ph…
• BeginnerTwo isolated conducting spheres, Sphere 1 of radius \(R\) and Sphere 2 of radius \(2R\), are connected by a long, thin conducting wire. A total net ch…
• BeginnerAn isolated, charged parallel-plate capacitor has a uniform electric field between its plates. A dielectric slab composed of polar molecules is insert…
• IntermediateA group of students designs an experiment to determine the dielectric constant \(\kappa\) of an unknown liquid. They use a parallel-plate capacitor wi…
• IntermediateA parallel-plate capacitor with vacuum capacitance \(C_0\) is connected to a battery that maintains a constant potential difference \(V_0\). A dielect…
• IntermediateFour identical capacitors, each with capacitance \(C\), are assembled into four distinct network combinations: - Combination I: All four capacitors c…
• IntermediateTwo irregularly shaped metallic conductors, 1 and 2, are connected by a thin conducting wire and given a net positive charge. The system is allowed to…
• IntermediateAn uncharged, thick conducting spherical shell has inner radius \(a\) and outer radius \(b\). A positive point charge \(+q\) is placed inside the cavi…
• IntermediateA circuit contains an ideal battery supplying potential difference \(V_0\) connected to four capacitors. Capacitor \(C_1\) of capacitance \(2C\) is co…
• IntermediateTwo isolated conducting spheres, Sphere 1 of radius \(R\) and Sphere 2 of radius \(2R\), are located far apart in a vacuum. Sphere 1 initially carries…
• IntermediateA neutral solid conducting sphere is held at a fixed distance from a point charge \(+Q\). The sphere is observed to experience a net electrostatic for…
• IntermediateTwo solid conducting spheres, \(A\) and \(B\), with radii \(R\) and \(2R\) respectively, are separated by a distance much larger than \(2R\) and conne…
• IntermediateAn uncharged, solid conducting sphere of radius \(R\) is placed near a positive point charge \(+q\). The point charge is fixed at a distance \(d\) (wh…
• AdvancedA very long, nonconducting cylindrical shell of inner radius \(R\) and outer radius \(2R\) has a non-uniform volume charge density given by…
• AdvancedA long cylindrical capacitor of length \(L\) consists of a solid inner conducting cylinder of radius \(a\) and a concentric outer conducting cylindric…
• AdvancedAn uncharged, solid conducting body of arbitrary outer shape contains an empty, irregularly shaped internal cavity. A point charge \(+q\) is fixed at…
• AdvancedThree solid conducting spheres \(A\), \(B\), and \(C\) have radii \(R_A = R\), \(R_B = 2R\), and \(R_C = 3R\), respectively, and are located far from…
• AdvancedA non-linear dielectric material placed between parallel conducting plates has an electric displacement field magnitude given by…
• AdvancedA capacitor consists of two rectangular conducting plates of length \(L\) and width \(W\) tilted at a small angle \(\theta\) relative to one another…
• AdvancedA spherical capacitor consists of a solid conducting sphere of radius \(a\) surrounded by a concentric conducting spherical shell of inner radius…
• AdvancedA parallel-plate capacitor consists of two conducting plates, each of area \(A\), separated by a distance \(d\). The region between the plates is fill…
• AdvancedA solid conducting sphere of radius \(R\) carries a net positive charge \(Q\), resulting in a uniform surface charge density \(\sigma\). Because like…
• AdvancedA cylindrical capacitor of length \(L\) consists of a solid conducting inner cylinder of radius \(a\) and a thin conducting outer cylindrical shell of…

Unit 11 · Electric Circuits

• BeginnerConsider the multi-loop circuit shown in the figure, consisting of two ideal batteries with electromotive forces \(\varepsilon_1 = 2\varepsilon_0\) an…
• BeginnerThe electric current \(I\) passing through a semiconductor device as a function of time \(t\) is given by \(I(t) = bt^2 + I_0\), where…
• BeginnerA capacitor of capacitance \(C\) is initially charged to a charge \(Q_0\) and then connected in series with a resistor of resistance \(R\) at time…
• BeginnerAn experiment is conducted to measure the current \(I\) through a non-ohmic thermistor as a function of the potential difference \(V\) applied across…
• BeginnerA simple circuit consists of an ideal DC voltage source delivering a constant potential difference \(V_0\) connected across a variable resistor of res…
• BeginnerA central junction in an electric circuit connects five branches carrying current, as shown in the table. | Branch | Current Magnitude | Direction Re…
• BeginnerTwo cylindrical resistors, Resistor 1 and Resistor 2, are made of the same uniform material. Resistor 1 has length \(L_0\) and cross-sectional radius…
• BeginnerFive identical resistors, each of resistance \(R\), are arranged in a bridge network between terminals \(X\) and \(Y\), as shown in the diagram. Four…
• BeginnerThree identical resistors, each of resistance \(R\), are connected to form a closed triangular loop (Configuration 1). A second identical set of three…
• BeginnerFour identical resistors, each with resistance \(R\), are connected to an ideal battery of electromotive force \(\mathcal{E}\), as shown in the circui…
• IntermediateAn ideal battery of voltage \(\varepsilon\) is connected in series with a switch \(S\) and a resistor of resistance \(R\). This series combination is…
• IntermediateA student is designing an experiment to investigate the magnetic field produced by a long, straight vertical wire. The student has a variable DC power…
• IntermediateA student is investigating how the resistance of a metal wire changes with temperature. The relationship between the resistance \(R\) of the wire and…
• IntermediateA real power supply with constant electromotive force \(\mathcal{E}\) and internal resistance \(r\) is connected in series with a variable load resist…
• IntermediateA hollow cylindrical resistor of length \(L\), inner radius \(a\), and outer radius \(b\) is made of a material with uniform resistivity \(\rho\). In…
• IntermediateA resistor made of a material with uniform resistivity \(\rho\) is manufactured in the shape of a truncated cone of length \(L\). The two circular end…
• IntermediateA real battery with electromotive force \(\mathcal{E}\) and internal resistance \(r\) is connected to an ideal high-resistance digital voltmeter. When…
• IntermediateIn a direct current circuit, a main line carrying a constant current \(I_0\) reaches a junction that splits into three parallel branches, which rejoin…
• IntermediateAn ideal current source delivering a constant total current \(I_0\) is connected in parallel with an uncharged capacitor of capacitance \(C\) and a re…
• IntermediateA long cylindrical wire of radius \(R\) carries a non-uniform current density parallel to its central axis given by \(J(r) = C r^2\), where \(r\) is t…
• AdvancedA circuit is constructed with an ideal battery of emf \( \varepsilon \), a switch S, two resistors of resistances \( R_1 \) and \( R_2 \), and an init…
• AdvancedA capacitor of capacitance \(C\) has an initial charge \(Q_0\) on its top plate and \(-Q_0\) on its bottom plate. It is connected in a circuit with an…
• AdvancedA curved resistor of uniform resistivity \(\rho\) is formed from a section of a coaxial cylinder of height \(h\), inner radius \(a\), and outer radius…
• AdvancedAt a junction node \(N\) in a circuit, current \(I_1\) flows into the node through branch 1, and current \(I_2\) flows out of the node through branch…
• AdvancedA non-uniform cylindrical resistor of length \(L\) has a uniform resistivity \(\rho_0\). The cross-sectional area \(A\) of the resistor varies with po…
• AdvancedA circuit contains an ideal battery of emf \(\mathcal{E}\), a switch \(S\), three identical resistors of resistance \(R_1 = R_2 = R_3 = R\), and an in…
• AdvancedA single closed circuit consists of an ideal battery with electromotive force \(\mathcal{E}\), a linear resistor of resistance \(R\), and a non-ohmic…
• AdvancedTwo concentric, thin conducting spherical shells have inner radius \(a\) and outer radius \(b\), respectively. The region between the shells (…
• AdvancedA liquid conductor with uniform resistivity \(\rho\) flows through a tapered pipe of length \(L\). The pipe has a circular cross-section whose radius…
• AdvancedA circuit contains an ideal battery of voltage \(V_0\) connected in series with resistor \(R_A\). Downstream of \(R_A\), the circuit splits into two p…

Unit 12 · Magnetic Fields and Electromagnetism

• BeginnerA very long, straight wire carries a constant current \(I\). At a radial distance \(r\) from the central axis of the wire, the magnitude of the magnet…
• BeginnerA positive ion of mass \(m\) and charge \(+q\) is injected with an initial horizontal speed \(v_0\) into a region of space containing a uniform electr…
• BeginnerA cylindrical bar magnet has a north pole at end \(N\) and a south pole at end \(S\). Which of the following correctly identifies the direction of the…
• BeginnerTwo long, thin, parallel wires separated by a distance \(d\) each carry a current \(I\) in the same direction, resulting in an attractive magnetic for…
• BeginnerTwo long, straight parallel wires in the plane of the page are separated by a distance \(2d\). Wire 1 carries a current \(I\) directed toward the top…
• BeginnerA thin circular wire loop of radius \(R\) lies in the \(xy\)-plane and carries a steady current \(I\). The magnitude of the magnetic field along the c…
• BeginnerA cyclotron with a uniform magnetic field \(B\) and maximum dee radius \(R\) is initially calibrated to accelerate protons of charge \(+e\) and mass \…
• BeginnerA velocity selector consists of a region with a uniform electric field \(\vec{E}\) directed upward and a uniform magnetic field \(\vec{B}\) directed i…
• BeginnerA particle with mass \(m\) and positive charge \(q\) travels with constant speed \(v\) in the \(+x\)-direction. At time \(t = 0\), the particle enters…
• BeginnerA rectangular loop of wire carrying a constant clockwise current \(I\) lies flat in the plane of the page. A uniform magnetic field \(\vec{B}\) is dir…
• IntermediateA student is designing an experiment to investigate the magnetic field produced by a long, straight vertical wire. The student has a variable DC power…
• IntermediateA particle of mass \(m\) and positive charge \(q\) is launched into a region containing a uniform magnetic field \(\vec{B}\) directed along the \(+z\)…
• IntermediateA long, hollow cylindrical conductor has an inner radius \(a\) and an outer radius \(b\). A total current \(I\) is uniformly distributed over the cros…
• IntermediateA rigid, planar circular loop of wire carrying a constant steady current \(I\) is placed in an external non-uniform magnetic field \(\vec{B}\). The lo…
• IntermediateA straight wire of length \(L\) lies along the \(x\)-axis from \(x = 0\) to \(x = L\) and carries a constant current \(I\) in the \(+x\)-direction. Th…
• IntermediateA thin, infinitely wide flat sheet lying in the \(xy\)-plane carries a uniform surface current density \(\vec{K} = K\hat{i}\), where \(K\) is constant…
• IntermediateA long, straight wire carries a steady current directed out of the page. Two closed circular Amperian paths, labeled \(C_1\) and \(C_2\), are situated…
• IntermediateA long, thick cylindrical conducting tube of inner radius \(a\) and outer radius \(b\) carries a uniformly distributed total current \(I\) parallel to…
• IntermediateA straight wire segment of length \(L\) lies in the \(xy\)-plane with one end anchored at the origin. The wire carries a constant electric current…
• IntermediateElectrons in a cathode-ray device are accelerated from rest through an initial potential difference \(V_0\) before entering a region containing a unif…
• AdvancedA long coaxial cable consists of a solid inner cylindrical conductor of radius \(a\) and a thin, concentric outer conducting cylindrical shell of radi…
• AdvancedTwo long, straight, parallel wires carry steady currents \(I_1\) and \(I_2\) perpendicular to the \(xy\)-plane. An arbitrary closed Amperian loop…
• AdvancedA long, hollow cylindrical conductor has an inner radius \(a\) and an outer radius \(b\). A total current \(I\) is distributed uniformly across the co…
• AdvancedA long, solid cylindrical conductor of radius \(R\) carries a uniform current density \(J\) parallel to its central axis. A cylindrical cavity of radi…
• AdvancedA particle with mass \(m\) and positive charge \(q\) enters a uniform magnetic field \(\vec{B}\) with velocity \(\vec{v}\). The velocity vector makes…
• AdvancedA particle of mass \(m\) and positive charge \(q\) enters a uniform magnetic field \(\vec{B} = B_0 \hat{k}\) with an initial velocity \(\vec{v}_0\) di…
• AdvancedA long, flat copper ribbon of width \(w\) lies in the \(xy\)-plane, centered on the \(y\)-axis between \(x = -w/2\) and \(x = w/2\). The ribbon carrie…
• AdvancedIn a magnetism experiment, two single-turn planar loops—a square loop of side length \(s\) and a circular loop of radius \(R\)—are constructed such th…
• AdvancedA long, solid cylindrical conductor of radius \(R\) carries a non-uniform current directed along its central axis. The current density inside the cond…
• AdvancedA flat, tightly wound spiral coil lies in a single plane and consists of \(N\) turns of fine wire. The spiral has an inner radius \(a\) and an outer r…

Unit 13 · Electromagnetic Induction

• BeginnerA stationary circular wire loop with an area of \(0.50 \text{ m}^2\) is placed in a region of a uniform magnetic field \(\vec{B}\) that is oriented pe…
• BeginnerA rectangular wire loop of length \(L\), width \(w\), and total resistance \(R\) is placed in a uniform magnetic field of magnitude \(B\) that points…
• BeginnerAn ideal radio tuning circuit consists of an inductor with inductance \(L\) and a capacitor with capacitance \(C\), resulting in an oscillation freque…
• BeginnerAn ideal LC circuit consists of an inductor with constant inductance \(L\) and an air-filled parallel-plate capacitor with plate area \(A\) and initia…
• BeginnerAn inductor in a circuit has self-inductance \(L\) and carries an initial steady current \(I_0\), storing a total magnetic energy \(U_0\). If the curr…
• BeginnerAn ideal battery, a resistor of resistance \(R\), an air-core inductor, and an open switch are connected in series. A ferromagnetic core with permeabi…
• BeginnerThree direct-current circuits, labeled Circuit X, Circuit Y, and Circuit Z, each consist of an ideal inductor and a resistor connected in series. The…
• BeginnerA solid copper plate attached to a light rod swings as a pendulum, entering a region of uniform magnetic field directed perpendicular to the plane of…
• BeginnerAn ideal long cylindrical solenoid with length \(\ell\), circular cross section of radius \(r\), and total number of turns \(N\) has a self-inductance…
• BeginnerAn ideal inductor of inductance \(L\) operates under four distinct test conditions where the current \(I\) changes linearly with time during a given t…
• IntermediateA circuit is constructed with an ideal battery of emf \(\varepsilon\), two resistors with resistances \(R_1\) and \(R_2\), an ideal inductor of induct…
• IntermediateA circuit containing a resistor of resistance \(R\) and an inductor of inductance \(L\) initially carries a steady-state current \(I_0\). At time…
• IntermediateA long, straight wire carries a constant current \(I\). A rectangular loop with side lengths \(L\) parallel to the wire and \(w\) perpendicular to the…
• IntermediateA long, straight wire carrying a current \(I\) lies in the plane of a square conducting loop of side length \(a\). The side of the loop nearest to the…
• IntermediateA circuit consists of an ideal battery of electromotive force \(\mathcal{E}\), a resistor of resistance \(R\), an inductor of inductance \(L\), and an…
• IntermediateA horizontal conducting bar of length \(\ell\) and mass \(m\) slides vertically downward without friction along two parallel vertical conducting rails…
• IntermediateA flexible circular loop of wire with radius \(r\) and total resistance \(R\) lies in a plane perpendicular to a uniform magnetic field of magnitude \…
• IntermediateA closed surface \(S\) is situated in the non-uniform magnetic field produced by a nearby magnetic dipole, as shown. Magnetic field lines pass through…
• IntermediateA flat rectangular loop of wire with resistance \(R\), length \(L\) parallel to a long straight wire, and width \(w\) sits in the same plane as the wi…
• IntermediateA flat, stationary circular loop of wire of radius \(a\) is placed in a region of uniform magnetic field directed perpendicular to the plane of the lo…
• AdvancedA capacitor of capacitance \(C\) has an initial charge \(Q_0\) on its top plate and \(-Q_0\) on its bottom plate. It is connected in a circuit with an…
• AdvancedA long, straight wire carrying a steady current \(I\) lies in the same plane as a right-triangular loop. The side of the loop of height \(h\) is paral…
• AdvancedA circuit consists of an ideal battery of constant emf \(\mathcal{E}\), a resistor of resistance \(R\), and an inductor whose inductance \(L(t)\) vari…
• AdvancedA large, flat circular coil with \(N_1\) turns and radius \(R\) lies in a plane. A much smaller, flat circular coil with \(N_2\) turns and radius…
• AdvancedCircuit 1 consists of an ideal capacitor of capacitance \(C\) connected in series with two ideal inductors of inductance \(L\) and \(3L\), with no mut…
• AdvancedA small magnetic dipole with dipole moment \(\vec{\mu}\) is located at the center of a fixed, flat circular conducting loop lying in the \(xy\)-plane.…
• AdvancedA flat circular coil consists of \(N\) tightly wound turns of wire, has radius \(r\), and has total electrical resistance \(R\). The coil lies initial…
• AdvancedAn ideal LC circuit consists of an inductor of inductance \(L\) and a parallel-plate capacitor filled completely with a dielectric slab of dielectric…
• AdvancedAn ideal battery of electromotive force \(\mathcal{E}\) is connected in series at time \(t = 0\) with an ideal switch S, a resistor of resistance…
• AdvancedA long, ideal solenoid of radius \(R\) has \(n\) turns per unit length and carries a time-dependent current given by…