Midterm Study Guide: Electrostatics, Current & Resistance, Circuits, and Magnetic Fields

Electrostatics

Coulomb’s Law

Coulomb’s Law describes the force between two point charges. The force is a vector, acting along the line joining the charges, and follows an inverse-square law.

• Formula: $\vec{F}_{12} = k \frac{q_1 q_2}{r^2} \hat{r}$

• Constant: $k = 8.99 \times 10^9$ N·m2/C2

• Direction: Attractive if charges are opposite; repulsive if same sign.

• Key Skills:

  • Break forces into components

  • Add vectors correctly

  • Use symmetry arguments

  • Recognize inverse-square behavior (if $r$ doubles, $F$ becomes 1/4)

• Common Problems: Equilateral triangle of charges, charges on a line, finding zero-force locations, ratio questions.

Electric Field

The electric field is the force per unit charge at a point in space, produced by source charges. It is a vector field, independent of any test charge placed in the field.

• Formula: $\vec{E} = k \frac{q}{r^2} \hat{r}$

• Direction: Away from positive, toward negative charges.

• Superposition Principle: The net field is the vector sum of fields from all charges: $\vec{E}_{net} = \vec{E}_1 + \vec{E}_2 + \ldots$

• Common Questions: Direction of net field at midpoint, zero-field points, field of two opposite charges, comparing magnitudes at different distances.

• Important Distinction: Force depends on test charge; field does not.

Electric Potential

Electric potential is a scalar quantity representing the potential energy per unit charge at a point. It adds algebraically, not as a vector.

• Formula: $V = k \frac{q}{r}$

• Relationship to Field: The electric field is the negative gradient (derivative) of potential.

• Distance Dependence: Potential decreases as $1/r$; field decreases as $1/r^2$.

• Key Points:

  • Potential can be zero even if field is not, and vice versa.

  • Energy relationships:

    • $U = qV$

    • $\Delta U = -W$ (change in potential energy equals negative work done by the field)

• Applications: Energy conservation problems involving potential and kinetic energy.

Motion in an Electric Field

Charged particles in a uniform electric field experience a constant force, leading to constant acceleration (analogous to projectile motion).

• Force: $F = qE$

• Acceleration: $a = \frac{qE}{m}$

• Motion: Horizontal motion at constant velocity; vertical motion with constant acceleration.

• Example: Electron in a parallel-plate capacitor follows a parabolic trajectory.

Current & Resistance

Current

Electric current is the rate of flow of charge through a conductor.

• Definition: $I = \frac{dQ}{dt}$

• Microscopic View: $I = nqAv_d$

  • $n$ = charge carrier density

  • $q$ = charge of carrier

  • $A$ = cross-sectional area

  • $v_d$ = drift velocity

• Conceptual Points: Electrons move slowly; the signal propagates at nearly the speed of light.

Ohm’s Law

Ohm’s Law relates voltage, current, and resistance in ohmic materials (those with constant resistance).

• Formula: $V = IR$

• Graph: For ohmic materials, $V$ vs $I$ is a straight line.

Resistance Formula

The resistance of a conductor depends on its material, length, and cross-sectional area.

• Formula: $R = \rho \frac{L}{A}$

• Proportional Relationships:

  • Doubling length ($L$) doubles resistance ($R$).

  • Doubling area ($A$) halves resistance ($R$).

Power in Electric Circuits

Electric power is the rate at which energy is transferred or converted in a circuit.

• Formulas:

  • $P = IV$

  • $P = I^2R$

  • $P = \frac{V^2}{R}$

• Application: Choose the formula based on known quantities.

Circuits

Series Circuits

In a series circuit, components are connected end-to-end, so the same current flows through each.

• Current: Same through all elements.

• Voltage: Splits among components.

• Total Resistance: Adds directly: $R_{eq} = R_1 + R_2 + \ldots$

• Voltage Divider Rule: $V_i = IR_i$

Parallel Circuits

In a parallel circuit, components are connected across the same two points, so the same voltage is applied to each.

• Voltage: Same across all branches.

• Current: Splits among branches.

• Total Resistance: Inverse sum: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$

• Key Point: Adding parallel resistors always reduces total resistance.

Kirchhoff’s Rules

Kirchhoff’s rules are used to analyze complex circuits with multiple loops and junctions.

• Junction Rule: The sum of currents entering a junction equals the sum leaving (conservation of charge).

• Loop Rule: The sum of voltage changes around any closed loop is zero (conservation of energy).

• Procedure:

  1. Assign current directions.

  2. Apply the junction rule at nodes.

  3. Write loop equations for each independent loop.

  4. Solve the resulting system of equations.

• Common Errors: Incorrect sign conventions, forgetting battery polarity, neglecting internal resistance.

Capacitors (If Covered)

Capacitors store energy in the electric field between their plates.

• Energy Stored: $U = \frac{1}{2}CV^2$

• Capacitance: $C = \frac{\varepsilon_0 A}{d}$

Magnetic Fields

Magnetic Field from a Long Straight Wire

A current-carrying wire produces a magnetic field that circles the wire, with magnitude decreasing with distance.

• Formula: $B = \frac{\mu_0 I}{2\pi r}$

• Direction: Determined by the right-hand rule (thumb in direction of current, fingers curl in direction of $B$).

Magnetic Force on a Moving Charge

A moving charge in a magnetic field experiences a force perpendicular to both its velocity and the field.

• Formula: $F = qvB\sin\theta$

• Key Properties:

  • Force is zero if velocity is parallel to $B$ ($\theta = 0$ or $180^\circ$).

  • Force is maximum if velocity is perpendicular ($\theta = 90^\circ$).

  • Magnetic force does no work (does not change speed, only direction).

Circular Motion in a Magnetic Field

Charged particles moving perpendicular to a uniform magnetic field undergo uniform circular motion.

• Force Balance: $qvB = \frac{mv^2}{r}$

• Radius: $r = \frac{mv}{qB}$

• Frequency: $f = \frac{qB}{2\pi m}$

• Application: Cyclotrons, mass spectrometers.

Force on a Current-Carrying Wire

A wire carrying current in a magnetic field experiences a force.

• Formula: $F = ILB\sin\theta$

• Direction: Right-hand rule (thumb: current, fingers: $B$, palm: force).