Difficulty: Intermediate | Prerequisites: Electrostatics and Gauss's Law study notes, basic circuit concepts (voltage, current, resistance)
This set of notes covers three areas that tend to cluster together on exams: capacitors (with and without dielectrics), transient behaviour in DC circuits containing resistors with capacitors or inductors, and the magnetic force on moving charges and current-carrying wires. These topics bridge the gap between static charges and full electromagnetic theory. If you are comfortable with electric fields and potentials, you are ready for this material.
Capacitors store charge and energy; inserting a dielectric multiplies the capacitance, and whether voltage or charge stays fixed depends on whether the battery remains connected. RC and RL circuits have exponential transient behaviour governed by time constants. Moving charges in magnetic fields experience a force perpendicular to both their velocity and the field, which is why charged particles travel in circles and why parallel current-carrying wires attract or repel.
Capacitance (C)
The ratio of the charge stored on a capacitor to the voltage across it, C = Q/V, measured in farads (F). Think of it as: how much charge the capacitor can hold per volt applied.
Dielectric constant (κ)
A dimensionless number (always ≥ 1) describing how much a dielectric material increases the capacitance of a capacitor. In simple terms, inserting a dielectric multiplies the capacitance by κ.
Time constant (τ)
The characteristic time for an exponential process in a circuit. For an RC circuit, τ = RC. For an RL circuit, τ = L/R. Think of it as: the time it takes for the current or voltage to reach about 63% of its final value (or decay to about 37%).
Self-inductance (L)
The property of a coil or circuit element that opposes changes in current through it, by generating an induced EMF. Measured in henrys (H). Think of it as: electrical inertia.
Magnetic force (Lorentz force)
The force on a charged particle moving through a magnetic field: F = qv x B. The force is always perpendicular to both the velocity and the field, so it changes the direction of motion but not the speed.
Cyclotron radius (R)
The radius of the circular path followed by a charged particle moving perpendicular to a uniform magnetic field. In simple terms, faster or heavier particles orbit in larger circles; stronger fields produce tighter circles.
A parallel-plate capacitor connected to a battery maintains a fixed voltage V across its plates.
If a dielectric with constant κ is inserted while the battery stays connected, the capacitance increases to κC.
Because V is fixed (battery enforces it), Q = CV increases by a factor of κ. The charge increases.
If the battery is disconnected first, the charge Q stays fixed and the voltage drops to V/κ instead.
This battery-connected vs. battery-disconnected distinction is one of the most common exam traps.
For capacitors in series, the reciprocals add: 1/C_eq = 1/C₁ + 1/C₂.
Two identical 6 μF capacitors in series: 1/C_eq = 1/6 + 1/6 = 2/6, so C_eq = 3 μF.
Series capacitance is always less than the smallest individual capacitor.
When a battery charges a capacitor through a resistor, the current starts at I₀ = V/R and decays exponentially.
After a very long time (t >> τ = RC), the capacitor is fully charged, the voltage across it equals the battery voltage, and the current drops to zero.
The current at any time: I(t) = (V/R) e^(–t/RC).
When a switch is closed on an RL circuit (inductor + resistor + battery), the inductor initially opposes any change in current. At t = 0, the current is zero.
The current rises exponentially toward its steady-state value V/R with time constant τ = L/R.
I(t) = (V/R)(1 – e^(–Rt/L)) during charging.
When the circuit reaches steady state and the switch is then opened at t = 0, the current decays: I(t) = (V/R) e^(–Rt/L). The inductor drives the current as it releases its stored energy.
A charge q moving with velocity v in a magnetic field B experiences a force F = qv x B.
For an electron moving in +x with B in +y: first compute v x B using the right-hand rule. x̂ x ŷ = +ẑ, so v x B points in +z. Then apply the electron's charge: F = q(v x B) = (–e)(vB)(+ẑ) = –evB ẑ. The force is in the –z direction. (Note: the exam's listed answer is +z. Work through the cross product carefully with the sign of the charge. The key step students miss is applying the negative sign of the electron after computing the cross product.)
Two parallel wires carrying current in the same direction attract each other.
Two parallel wires carrying current in opposite directions repel each other.
The force per unit length is F/L = μ₀ I₁ I₂ / (2πd), where d is the separation.
The inductance of a solenoid: L = μ₀ N² A / ℓ, where N is the number of turns, A is the cross-sectional area, and ℓ is the length.
L is proportional to N². Doubling N (while keeping length and area constant) quadruples L.
A particle of charge q and mass m, accelerated through a potential difference V, gains kinetic energy: qV = ½mv², so v = √(2qV/m).
Entering a uniform magnetic field B perpendicular to its velocity, it follows a circular path.
The magnetic force provides the centripetal force: qvB = mv²/r.
Solving for the radius: R = mv/(qB) = (1/B)√(2mV/q).
C = \kappa C_0 = \kappa \frac{\varepsilon_0 A}{d}Capacitance of a parallel-plate capacitor with a dielectric.
\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2}Capacitors in series. The equivalent capacitance is always smaller than the smallest.
I(t) = \frac{V}{R} e^{-t/RC}Current in an RC circuit during charging. The time constant is τ = RC.
I(t) = \frac{V}{R}\left(1 - e^{-Rt/L}\right)Current in an RL circuit after the switch is closed (charging). Time constant τ = L/R.
I(t) = \frac{V}{R} e^{-Rt/L}Current in an RL circuit after the switch is opened (discharging). The current decays from V/R toward zero.
\vec{F} = q\vec{v} \times \vec{B}Magnetic (Lorentz) force on a moving charge. The force is perpendicular to both v and B.
R = \frac{mv}{qB} = \frac{1}{B}\sqrt{\frac{2mV}{q}}Cyclotron radius for a particle of mass m and charge q accelerated through potential V into field B.
L = \mu_0 \frac{N^2 A}{\ell}Self-inductance of a solenoid. Proportional to N², so doubling the turns quadruples L.
Students often think that inserting a dielectric always increases the stored charge. It depends on whether the battery is connected. Battery connected: Q increases. Battery disconnected: Q stays the same, voltage drops.
Students confuse series and parallel capacitor formulas with resistor formulas. Capacitors in series add reciprocally (like resistors in parallel). Capacitors in parallel add directly (like resistors in series).
Students assume that in an RL circuit, the current jumps immediately to V/R when the switch closes. The inductor prevents this; current starts at zero and rises gradually.
Students forget to account for the sign of the charge when applying the right-hand rule. For an electron, find v x B with the right-hand rule first, then reverse the direction.
RC circuits are everywhere: the timing circuit in a camera flash uses the RC time constant to control how long the flash charges before firing. Inductors in RL circuits are used in power supplies to smooth current fluctuations. The cyclotron radius principle is the basis of mass spectrometers, which separate ions by mass, and particle accelerators like cyclotrons used in medical imaging.
⚠️ The dielectric question almost always specifies whether the battery stays connected. Read this carefully; it changes the answer completely.
⚠️ RC circuit long-time behaviour: current goes to zero, capacitor fully charged. RL circuit at t = 0: current is zero, inductor opposes change. These limiting cases are quick marks.
⚠️ Solenoid inductance is proportional to N². If N doubles, L quadruples. This is a common numerical question.
⚠️ The cyclotron radius derivation (combining energy conservation from the potential difference with the centripetal force condition) is a classic short-answer or long-answer problem.
True or False: When a dielectric is inserted into a capacitor while the battery remains connected, the voltage across the capacitor increases. (False. The battery holds the voltage constant; the charge increases.)
Fill in the blank: The equivalent capacitance of two 6 μF capacitors in series is __________ μF. (3)
True or False: In an RC circuit being charged, the current after a very long time approaches V/R. (False. It approaches zero.)
Fill in the blank: Doubling the number of turns in a solenoid (constant length and area) multiplies the inductance by __________. (4)
True or False: Two parallel wires carrying current in the same direction repel each other. (False. They attract.)
Q: A parallel-plate capacitor is connected to a battery with voltage V. A dielectric with constant κ is inserted while the battery remains connected. What happens to the stored charge Q?
A: Q increases by a factor of κ. The battery fixes V, capacitance increases to κC, and Q = CV becomes κCV.
Q: What is the current in an RC circuit after a very long time (t >> τ)?
A: Zero. The capacitor is fully charged, no current flows.
Q: In an RL circuit, what is the current immediately after the switch is closed?
A: Zero. The inductor opposes any instantaneous change in current. The current then rises gradually toward V/R.
Q: An inductor L and resistor R are in series with battery V. The circuit reaches steady state, then the switch is opened at t = 0. What is I(t) for t > 0?
A: I(t) = (V/R) e^(–Rt/L). The current decays exponentially from its steady-state value V/R with time constant τ = L/R.
Q: A particle of charge q and mass m is accelerated from rest through potential V, then enters a magnetic field B perpendicular to its velocity. Derive the radius R of its circular path.
A: Energy conservation gives qV = ½mv², so v = √(2qV/m). The magnetic force provides centripetal acceleration: qvB = mv²/r. Solving: R = mv/(qB) = (1/B)√(2mV/q).
Q: How does the self-inductance of a solenoid change if the number of turns is doubled (length and area constant)?
A: It quadruples. L = μ₀ N² A / ℓ, so L is proportional to N².
Capacitance connects directly to the energy stored in electric fields (U = ½CV²), which you will need for LC oscillation circuits. RC and RL circuits are the building blocks for understanding RLC circuits and AC circuit analysis. The magnetic force on moving charges is the foundation for understanding how motors, generators, and cyclotrons work, and it links to Faraday's Law once you consider what happens when charges move through changing magnetic fields.
Capacitance, dielectric, dielectric constant, parallel-plate capacitor, series capacitors, parallel capacitors, RC circuit, time constant, exponential decay, RL circuit, inductor, self-inductance, solenoid, magnetic force, Lorentz force, right-hand rule, cross product, cyclotron radius, mass spectrometer, parallel wires, current-carrying wires, mutual attraction, henry, farad, PHYS 212, University Physics E&M