Electricity and Magnetism Mid-Term Practice Guide

🟢 Topic 1: Discrete Point Charges (Electrostatics & Superposition)

Found in: Exam 1 (Ex 1 a-c), Exam 2 (Ex 1), Exam 3 (Ex 1 a-f), Exam 4 (Ex 1 a-c)
Question Types:

• Finding the net Coulomb force $\overrightarrow{F}$ or Electric field $\overrightarrow{E}$ at a specific point.
• Finding the Electric Potential $V$ at a specific point.
• Finding the Potential Energy $U$ of a system or the work $W$ required to bring in a new charge.
• "Nullifying" the field/force by placing a new charge, or finding equilibrium positions.

Formulas to use (from formula sheet):

• Force: $\overrightarrow{F}=k\frac{q_a{q}_b}{|{\overrightarrow{r}}_b-{\overrightarrow{r}}_a{|}^3}({\overrightarrow{r}}_b-{\overrightarrow{r}}_a)$
• Electric Field: $\overrightarrow{E}=k\sum \frac{q_i}{|\overrightarrow{r}-{\overrightarrow{r}}_i{|}^3}(\overrightarrow{r}-{\overrightarrow{r}}_i)$
• Potential: $V=k\sum \frac{q_i}{|\overrightarrow{r}-{\overrightarrow{r}}_i|}$
• Work/Energy: $W=\mathrm{\Delta }U$, $U_{tot}=\frac{1}{2}\sum k\frac{q_i{q}_j}{|{\overrightarrow{r}}_j-{\overrightarrow{r}}_i|}$

🛠️ Step-by-Step Solution Plan:

1. Define coordinates: Write down the exact $(x,y,z)$ coordinates of every charge and the observation point.
2. Calculate distance vectors: Use the formula sheet's definition: ${\overrightarrow{r}}_{ab}={\overrightarrow{r}}_{obs}-{\overrightarrow{r}}_{source}$.
   • Calculate the magnitude $|{\overrightarrow{r}}_{ab}|=\sqrt{x^2+y^2+z^2}$ separately.
3. Apply Superposition: Plug the vectors and magnitudes directly into the summation formulas. Keep the components ($\hat{x},\hat{y},\hat{z}$) separate until the very end.
4. For Equilibrium / Nullifying: Set your resultant $\overrightarrow{E}$ or $\overrightarrow{F}$ equation to zero. If you need to cancel a specific component (e.g., $E_y=0$), isolate the $\hat{y}$ terms, equate them to zero, and solve for the unknown charge $Q$ or distance.
5. For Potential/Work: Remember that $V$ is a scalar. Do not use vectors here, just the magnitudes of the distances.

🟡 Topic 2: Gauss's Law & Highly Symmetric Geometries

Found in: Exam 1 (Ex 2), Exam 2 (Ex 2), Exam 3 (Ex 2 a-d), Exam 4 (Ex 2)
Question Types:

• Finding $\overrightarrow{E}$ inside/outside thick spherical shells, massive cylinders, or infinite slabs with uniform or non-uniform charge density $\rho (r)$ or $\rho (x)$.
• Evaluating configurations with empty cavities (eccentric cavities).
• Calculating electric flux $\mathrm{\Phi }$.

Formulas to use (from formula sheet):

• Gauss's Law: ${\oint }_S\overrightarrow{E}\cdot d\overrightarrow{A}=\frac{q_{enclosed}}{{\epsilon }_0}$
• Volume enclosed: $q_{enclosed}={\int }_{V_s}\rho ({\overrightarrow{r}}^{\prime })dV$
• Mathematics Section: $d{V}_{spherical}=4\pi r^{2}dr$, $d{V}_{cylindrical}=2\pi rLdr$, $d{V}_{cartesian}=Adx$

🛠️ Step-by-Step Solution Plan:

1. Identify the Symmetry & Gaussian Surface:
   • Spherical: Use a concentric sphere. Area $=4\pi r^2$.
   • Cylindrical: Use a coaxial cylinder. Area $=2\pi rL$.
   • Planar (Slab/Sheet): Use a pillbox/cylinder spanning across the surface. Area $=2A$.
2. Calculate $Q_{enclosed}$:
   • If density is constant: $Q_{enc}=\rho \times V_{enclosed}$.
   • If density varies (e.g., $\rho (r)={\rho }_0\frac{r}{a}$), set up the integral: $Q_{enc}=\int \rho (r)dV$ using the proper $dV$ from your math cheat sheet. Integrate from the start of the charge up to your observation radius $r$.
3. Apply Gauss's Law: Set $E\cdot (\text{Area})=\frac{Q_{enc}}{{\epsilon }_0}$ and solve for $E$.
4. The "Cavity" Trick (Superposition): If a problem features an off-center cavity, treat it as two solid objects superimposed:
   • Object 1: A solid shape with positive density $+\rho$.
   • Object 2: A smaller solid shape (the cavity) with negative density $-\rho$.
   • Calculate ${\overrightarrow{E}}_{full}+{\overrightarrow{E}}_{cavity}$ using standard Gauss's law for each.

🔵 Topic 3: Direct Integration of Continuous Charges

Found in: Exam 1 (Ex 1 d-f), Exam 4 (Ex 1 d-f)
Question Types:

• Finding $\overrightarrow{E}$ or $V$ for finite rods, rings, disks, or planes with holes.

Formulas to use (from formula sheet):

• $\overrightarrow{E}({\overrightarrow{r}}_a)=k\int \frac{\lambda ({\overrightarrow{r}}^{\prime })}{|{\overrightarrow{r}}_a-{\overrightarrow{r}}^{\prime }{|}^3}({\overrightarrow{r}}_a-{\overrightarrow{r}}^{\prime })dl$
• Integration tables in the "Mathematics" section.

🛠️ Step-by-Step Solution Plan:

1. Draw and establish $dq$: Express the charge element.
   • 1D (rod): $dq=\lambda dx$
   • 2D (disk/hole): $dq=\sigma dA=\sigma (2\pi rdr)$ or $\sigma (\rho d\rho d\phi )$.
2. Vector mapping: Write the vector pointing from the charge element to your observation point. Determine the magnitude (usually involves a Pythagorean form like $\sqrt{z^2+r^2}$).
3. Exploit Symmetry: Before integrating, explicitly state which components cancel out (e.g., "due to rotational symmetry, $E_x=E_y=0$, only $E_z$ remains").
4. Set up the integral: Substitute your $dq$ and vectors into $d\overrightarrow{E}=\frac{kdq}{r^2}\hat{r}$.
5. Evaluate: Match your integral to the "Table of Integrals" on the last page of your formula sheet (e.g., $\int \frac{xdx}{(x^2\pm a^2{)}^{3/2}}$).

🟣 Topic 4: Capacitors & Dielectrics

Found in: Exam 1 (Ex 3), Exam 2 (Ex 3), Exam 3 (Ex 3), Exam 4 (Ex 3)
Question Types:

• Calculating capacitance $C$ of cylindrical or spherical capacitors from scratch.
• Adding dielectrics (${\epsilon }_r$) or spacers.
• Calculating stored energy $U$ or max breakdown voltage.

Formulas to use (from formula sheet):

• $\mathrm{\Delta }{V}_{ab}=-{\int }_a^b\overrightarrow{E}\cdot d\overrightarrow{l}$
• $C=\frac{Q}{|\mathrm{\Delta }V|}$
• $U=\frac{C{V}^2}{2}=\frac{Q^2}{2C}$

🛠️ Step-by-Step Solution Plan:

1. Find the E-field first: Use Gauss's law to find $\overrightarrow{E}$ in the gap between the conductors. Assume the inner conductor has charge $+Q$.
   • Note on dielectrics: If the gap has a dielectric, replace ${\epsilon }_0$ with ${\epsilon }_0{\epsilon }_r$ in your Gauss's Law denominator.
2. Integrate to find $\mathrm{\Delta }V$: Evaluate $\mathrm{\Delta }V={\int }_{R_{inner}}^{R_{outer}}E(r)dr$. (Take the absolute value to ensure capacitance is positive).
3. Divide out $Q$: Plug $\mathrm{\Delta }V$ into $C=Q/\mathrm{\Delta }V$. The $Q$ will perfectly cancel out, leaving you with an expression based only on geometry ($R_{in},R_{out},L$) and $\epsilon$.
4. Handling Spacers/Layers (Equivalent Circuits):
   • If a dielectric is layered side-by-side longitudinally (like spacers along a cylinder), treat them as Capacitors in Parallel ($C_{tot}=C_1+C_2$).
   • If dielectrics are layered radially (inner shell, then outer shell), treat them as Capacitors in Series ($\frac{1}{C_{tot}}=\frac{1}{C_1}+\frac{1}{C_2}$).

🟠 Topic 5: Current Density, Resistance, & Ohm's Law

Found in: Exam 1 (Ex 4), Exam 2 (Ex 4), Exam 3 (Ex 4), Exam 4 (Ex 4)
Question Types:

• Finding Drift Velocity $v_d$ or charge carrier density $n$.
• Calculating resistance $R$ for non-uniform macroscopic objects via integration.
• Skin-effect (high frequency) where current density $J(r)$ varies radially.

Formulas to use (from formula sheet):

• $J=\frac{I}{A}$
• $J=nq{v}_d$ (Microscopic Ohm's Law)
• $\overrightarrow{J}=\sigma \overrightarrow{E}$
• $R=\frac{\rho L}{A}$

🛠️ Step-by-Step Solution Plan:

Scenario A: Drift Velocity and Charge Density

1. Find $J$ by dividing total Current $I$ by cross-sectional Area $A$.
2. If asked for $n$ (electrons per unit volume), use mass density and molar mass: $n={\rho }_{mass}\times \frac{N_A}{m_{atomic}}$. (Crucial pitfall: Convert molar mass from g/mol to kg/mol!)
3. Use $v_d=\frac{J}{nq}$ to find drift velocity. If there are positive and negative ions moving (electrolysis), remember $J=(n_1{q}_1{v}_{d1})+(n_2{q}_2{v}_{d2})$.

Scenario B: Calculating Non-Uniform Resistance (Integration)

1. Determine the direction of current flow.
2. Current flows Radially (e.g., from an inner cylinder to an outer ring):
   • The "resistors" are thin concentric cylindrical shells in series.
   • Write $dR=\frac{\rho (r)dr}{Area(r)}=\frac{\rho (r)dr}{2\pi rL}$.
   • Integrate $R={\int }_{R_{inner}}^{R_{outer}}dR$.
3. Current flows Longitudinally, but $J$ varies (Skin Effect):
   • The current isn't evenly distributed.
   • Use $I=\int J(r)dA$. Because it's a cross-section of a wire, $dA=2\pi rdr$.
   • Integrate $I={\int }_0^{R}J(r)2\pi rdr$ using the integral sheet.
   • Once you have the new $I$, find Power using $P=I^2{R}_{DC}$.

🚨 Golden Rules based on Your Professor's Grading Notes:

1. Always specify units: You will lose points if you just write a number. The prompt explicitly says "Simply stating numerical results will yield no points" and "Fill in the measure units for the final results".
2. Vectors need hats: If the prompt says "a vector quantity is requested", your final answer must include $\hat{x},\hat{y},\hat{z}$ or $\hat{r}$, $\hat{\phi }$.
3. Reference Potentials: For point charges and finite objects, $V(\mathrm{\infty })=0$. But for infinite objects (like infinite slabs or long cylinders), $V(\mathrm{\infty })$ blows up. You must choose a finite location (like $r=0$ or $x=0$) as your $V=0$ reference point. Justify this with one sentence if asked.