Electromagnetism — CheriMathLab

Key Concepts & Laws

The unified framework connecting electricity, magnetism, and light.

Coulomb's Law

Force between point charges: $$F = \frac{kq_1 q_2}{r^2}$$ where $k = 8.99 \times 10^9\;\text{N}\!\cdot\!\text{m}^2/\text{C}^2$

Electric Field & Potential

Field: $$\vec{E} = \frac{\vec{F}}{q} = \frac{kQ}{r^2}\,\hat{r}$$ Potential: $V = \frac{kQ}{r}$
Relation: $\vec{E} = -\frac{dV}{dr}\,\hat{r}$

Gauss's Law

$$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$ Powerful for symmetric charge distributions: spheres, cylinders, infinite planes.

Capacitance

$C = \frac{Q}{V} = \frac{\varepsilon_0 A}{d}$ (parallel plate)
Energy: $U = \tfrac{1}{2}CV^2 = \frac{Q^2}{2C}$
Series: $\frac{1}{C} = \sum \frac{1}{C_i}$ Parallel: $C = \sum C_i$

Ohm's Law & Circuits

$V = IR$ $P = IV = I^2 R$
Kirchhoff's junction rule: $\sum I_{\text{in}} = \sum I_{\text{out}}$
Kirchhoff's loop rule: $\sum V = 0$

Magnetic Force & Field

Force on charge: $$\vec{F} = q\vec{v} \times \vec{B}$$ Force on wire: $\vec{F} = I\vec{L} \times \vec{B}$
Biot-Savart: $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l} \times \hat{r}}{r^2}$

Faraday's Law

Induced EMF: $$\mathcal{E} = -\frac{d\Phi_B}{dt}$$ Magnetic flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$
Lenz's law: induced current opposes the change.

Maxwell's Equations

$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$ $$\nabla \cdot \vec{B} = 0$$ $$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$ $$\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$$

Problems

12 problems from electrostatics to electromagnetic waves.

Easy

Two point charges, $q_1 = +3\;\mu\text{C}$ and $q_2 = -5\;\mu\text{C}$, are separated by 20 cm. (a) Find the magnitude and direction of the electric force on each charge. (b) At what point along the line joining them is the net electric field zero?

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Use Coulomb's law directly. The field-zero point lies outside the charges, beyond the smaller charge. Set $\frac{kq_1}{r_1^2} = \frac{kq_2}{r_2^2}$ and solve.

Medium

A thin ring of radius 15 cm carries a total charge of $+8\;\mu\text{C}$ uniformly distributed. Find the electric field on the axis of the ring at a distance of 20 cm from its center. At what distance from the center is the field maximum?

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On-axis field of a ring: $E = \frac{kQx}{(x^2 + R^2)^{3/2}}$. To find the maximum, take $dE/dx = 0$, yielding $x = R/\sqrt{2}$.

Medium

Use Gauss's law to find the electric field inside and outside a solid non-conducting sphere of radius $R = 10\;\text{cm}$ carrying a uniform volume charge density $\rho = 5 \times 10^{-6}\;\text{C/m}^3$. Plot $E$ as a function of $r$.

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Choose a spherical Gaussian surface. Inside ($r < R$): $E = \frac{\rho r}{3\varepsilon_0}$. Outside ($r > R$): $E = \frac{\rho R^3}{3\varepsilon_0 r^2}$. $E$ is continuous at $r = R$.

Medium

Three capacitors ($C_1 = 2\;\mu\text{F}$, $C_2 = 4\;\mu\text{F}$, $C_3 = 6\;\mu\text{F}$) are connected: $C_1$ in series with the parallel combination of $C_2$ and $C_3$. A 12 V battery is connected across the combination. Find the charge and voltage across each capacitor, and the total energy stored.

Show Hint

$C_2$ and $C_3$ in parallel: $C_p = 10\;\mu\text{F}$. Then $C_1$ in series with $C_p$: $\frac{1}{C_{\text{total}}} = \frac{1}{2} + \frac{1}{10}$. Charge on series capacitors is equal. Voltage divides inversely to capacitance.

Easy

A copper wire (resistivity $1.68 \times 10^{-8}\;\Omega\!\cdot\!\text{m}$) has a diameter of 1.0 mm and length 50 m. (a) Find the resistance of the wire. (b) If a 9 V battery drives current through it, find the current and power dissipated. (c) How much energy is dissipated as heat in one hour?

Show Hint

$R = \frac{\rho L}{A}$ where $A = \pi(d/2)^2$. Then $I = V/R$ and $P = I^2 R$. Energy $= P \times t$.

Hard

A proton ($m = 1.67 \times 10^{-27}\;\text{kg}$) enters a region of uniform magnetic field $B = 0.5\;\text{T}$ perpendicular to its velocity of $3 \times 10^6\;\text{m/s}$. (a) Find the radius of the circular orbit. (b) Find the period of revolution. (c) If an electric field is also present perpendicular to both $\vec{v}$ and $\vec{B}$, what magnitude $E$ would make the proton travel in a straight line?

Show Hint

Radius: $r = \frac{mv}{qB}$. Period: $T = \frac{2\pi m}{qB}$ (independent of speed!). For straight-line motion, $qE = qvB$, so $E = vB$ — this is a velocity selector.

Hard

Two long, parallel wires carry currents $I_1 = 10\;\text{A}$ and $I_2 = 6\;\text{A}$ in opposite directions, separated by 8 cm. (a) Find the magnetic field at a point midway between the wires. (b) At what point between the wires is the net field zero? (c) Find the force per unit length between the wires.

Show Hint

Field from a long wire: $B = \frac{\mu_0 I}{2\pi r}$. Opposite currents means fields add at the midpoint. The zero-field point is closer to the weaker current. Force: $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$, repulsive for opposite currents.

Hard

A circular coil of 200 turns and radius 5 cm is placed in a uniform magnetic field of 0.3 T. The field decreases linearly to zero in 0.02 s. (a) Find the induced EMF. (b) If the coil has resistance $8\;\Omega$, find the induced current. (c) What total charge flows through the coil?

Show Hint

EMF: $\mathcal{E} = -N\frac{d\Phi}{dt} = -N \cdot A \cdot \frac{\Delta B}{\Delta t}$. Current: $I = \frac{\mathcal{E}}{R}$. Total charge: $Q = It = \frac{N\Delta\Phi}{R} = \frac{NBA}{R}$.

Medium

An RC circuit has $R = 10\;\text{k}\Omega$ and $C = 47\;\mu\text{F}$, connected to a 24 V battery. (a) Find the time constant. (b) Find the charge on the capacitor and the current at $t = 0.5\;\text{s}$. (c) How long does it take for the capacitor to reach 99% of its maximum charge?

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Time constant $\tau = RC$. Charging: $Q(t) = Q_{\max}(1 - e^{-t/\tau})$ and $I(t) = \frac{V}{R}e^{-t/\tau}$. For 99%: $1 - e^{-t/\tau} = 0.99$, so $t = \tau\ln(100) \approx 4.6\tau$.

Advanced

An electromagnetic wave in vacuum has an electric field amplitude $E_0 = 300\;\text{V/m}$. (a) Find the magnetic field amplitude. (b) Calculate the intensity (average power per unit area). (c) Find the radiation pressure on a perfectly reflecting surface. (d) How does this compare to atmospheric pressure?

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$B_0 = E_0/c$. Intensity: $I = \tfrac{1}{2}\varepsilon_0 c E_0^2$. Radiation pressure on a reflector: $P = 2I/c$. Atmospheric pressure is $\sim 10^5\;\text{Pa}$ — compare the magnitudes.

Advanced

A solenoid has 500 turns, length 25 cm, and cross-sectional area $4\;\text{cm}^2$. (a) Find its inductance. (b) If the current increases from 0 to 3 A in 0.01 s, find the induced back-EMF. (c) Find the energy stored when the current is 3 A. (d) Derive the energy density of the magnetic field inside.

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$L = \frac{\mu_0 N^2 A}{\ell}$. Back-EMF: $\mathcal{E} = -L\frac{dI}{dt}$. Energy: $U = \tfrac{1}{2}LI^2$. Energy density: $u = \frac{B^2}{2\mu_0}$ where $B = \mu_0 nI$.

Advanced

Thought Experiment: Maxwell added the displacement current term $\mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$ to Ampère's law. (a) Explain why this term is necessary for the consistency of the equations (consider charging a capacitor). (b) Show that Maxwell's equations predict electromagnetic waves traveling at speed $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$. (c) Calculate this speed and compare with the measured speed of light.

Show Hint

Without displacement current, $\nabla \cdot (\nabla \times \vec{B}) \neq 0$ during charging. To derive waves, take the curl of Faraday's law and substitute Ampère-Maxwell — you get a wave equation. Plugging in SI values: $c \approx 3 \times 10^8\;\text{m/s}$.