Course: University Physics 212: Electricity and Magnetism | University of Illinois at Urbana-Champaign
Source: Final Assessment review material
Difficulty: Intermediate Prerequisites: Electrostatics and Gauss's Law (Part 1 of these notes), basic calculus
Tags: electric potential, voltage, potential energy, equipotential surface, capacitance, dielectric, parallel plate capacitor, series capacitors, parallel capacitors, Ohm's law, resistance, Kirchhoff's rules, junction rule, loop rule, RC circuit, time constant, PHYS 212
Once you understand how charges create electric fields, the next question is: how much energy is involved in moving charges around? Electric potential and potential energy answer that question, and they lead directly into capacitors (devices that store energy in an electric field) and circuits (systems where charges flow continuously). This section bridges the "statics" of Part 1 with the "dynamics" of current flow. If you are comfortable with Coulomb's law and Gauss's law, you are ready for this material.
Electric potential is the energy landscape that charges move through. Capacitors store energy by holding charge at a voltage difference, and dielectrics increase their storage capacity. DC circuits obey Ohm's law and Kirchhoff's rules, which are statements of energy conservation and charge conservation. RC circuits describe how capacitors charge and discharge exponentially with a characteristic time constant τ = RC.
Electric potential (V)
The electric potential energy per unit charge at a point in space: V = U/q. In simple terms, it tells you how much "push" a charge would feel at that location, measured in volts (J/C).
Electric potential energy (U)
The energy stored in a configuration of charges due to their positions: U = kq₁q₂/r for two point charges. Think of it as the work you would need to do to assemble the charges from infinitely far apart.
Equipotential surface
A surface on which the electric potential is the same everywhere. No work is done moving a charge along an equipotential. These surfaces are always perpendicular to the electric field lines.
Capacitance (C)
The ratio of charge stored to voltage applied: C = Q/V. Measured in farads (F). A larger capacitance means more charge stored per volt.
Dielectric constant (κ)
A dimensionless number (κ > 1 for all real materials) that describes how much a dielectric material increases a capacitor's capacitance: C_new = κC₀. In simple terms, inserting a dielectric lets the capacitor store more energy at the same voltage.
Ohm's law
The voltage across a resistor equals the current through it times its resistance: V = IR. This is the fundamental relationship for resistive circuit elements.
Kirchhoff's junction rule
The sum of currents entering any junction equals the sum of currents leaving it. This is a direct statement of conservation of charge.
Kirchhoff's loop rule
The sum of all voltage changes around any closed loop in a circuit is zero. This is a statement of conservation of energy.
RC time constant (τ)
The product τ = RC sets the timescale for charging and discharging a capacitor through a resistor. After one time constant, a charging capacitor reaches about 63% of its maximum charge.
The relationship between electric field and potential is: E = −dV/dx (in one dimension). Where E is zero, V is not changing, so V must be constant.
A constant potential does not mean V = 0. It means V has the same value everywhere in that region. It could be 5 V, 100 V, or zero.
For two point charges q₁ and q₂ separated by distance r:
U = kq₁q₂/r
Like charges (both positive or both negative): U is positive (energy was put in to push them together).
Unlike charges (one positive, one negative): U is negative (energy is released as they come together).
If you replace one positive charge with a negative charge of the same magnitude, U changes sign. A system with potential energy U becomes −U.
When the electric field does positive work on a charge, it moves the charge in the direction of the force (lower potential energy).
A negative charge is attracted toward positive charges. Bringing a negative charge from infinity to a point between two positive charges means the field does positive work (the charge "falls" toward lower potential energy).
The change in potential energy is: ΔU = qΔV = q(V_B − V_A).
For an electron (q = −1.6 × 10⁻¹⁹ C) moved in the direction of the field through distance d in a uniform field E:
ΔV = −Ed (potential decreases in the direction of the field)
ΔU = qΔV = (−1.6 × 10⁻¹⁹)(−200 × 0.05) = +1.6 × 10⁻¹⁸ J
Positive ΔU means the electron's potential energy increased (you had to do work against the field to push the electron in the direction of E, since the field pushes electrons opposite to E).
Parallel-plate capacitor (no dielectric): C₀ = ε₀A/d, where A is plate area and d is plate separation.
With a dielectric filling the gap: C = κε₀A/d = κC₀.
Energy stored: U = ½CV².
When connected to a battery (constant V) and a dielectric is inserted:
C increases by factor κ.
V stays the same (battery enforces it).
U = ½CV² increases by factor κ. More energy is stored.
The battery supplies the additional charge needed.
When disconnected from the battery (constant Q) and a dielectric is inserted:
C increases by factor κ.
V decreases by factor κ (since V = Q/C).
U = Q²/(2C) decreases by factor κ. Energy is released.
Series: 1/C_eq = 1/C₁ + 1/C₂ + ...
For two capacitors: C_eq = C₁C₂/(C₁ + C₂)
Example: C₁ = 2 μF, C₂ = 3 μF → C_eq = (2)(3)/(2+3) = 1.2 μF
Series capacitance is always less than the smallest individual capacitor.
Parallel: C_eq = C₁ + C₂ + ...
Parallel capacitance is always larger than the largest individual capacitor.
Note: this is the opposite pattern to resistors (resistors in series add; capacitors in series use the reciprocal formula).
V = IR.
Resistors in parallel: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + ...
Three 30 Ω resistors in parallel: 1/R_eq = 3/30 = 1/10 → R_eq = 10 Ω.
Total current from a 10 V battery: I = V/R_eq = 10/10 = 1.0 A.
Resistors in series: R_eq = R₁ + R₂ + R₃ + ...
Electric field inside a wire: For a uniform wire of length L connected to a battery of voltage V, the field is E = V/L (assuming negligible internal resistance).
Junction rule (conservation of charge): ΣI_in = ΣI_out at every node.
Loop rule (conservation of energy): ΣΔV = 0 around any closed loop.
These two rules, combined with Ohm's law, let you solve any DC circuit.
Charging: q(t) = Q_max(1 − e^(−t/RC)), where Q_max = εC (ε is the battery EMF).
At t = τ = RC, the capacitor reaches about 63% of Q_max (since 1 − e⁻¹ ≈ 0.63).
At t = 5τ, the capacitor is more than 99% charged, effectively fully charged.
Discharging: q(t) = Q₀e^(−t/RC).
At t = τ, the charge has dropped to about 37% of Q₀ (since e⁻¹ ≈ 0.37).
At t = 2τ, q = Q₀e⁻² ≈ 0.135 Q₀.
The derivation of the charging equation starts from Kirchhoff's loop rule:
ε − IR − q/C = 0
Since I = dq/dt: ε − R(dq/dt) − q/C = 0
Rearranging and separating variables gives the exponential solution.
Quantity | Formula | Notes |
|---|---|---|
Point charge potential energy | U = kq₁q₂/r | Positive for like charges, negative for unlike |
Potential difference | ΔV = −∫E · dl | Or simply ΔV = −Ed for uniform fields |
Change in potential energy | ΔU = qΔV | Watch the sign of q |
Parallel-plate capacitance | C = κε₀A/d | κ = 1 for vacuum |
Energy in capacitor | U = ½CV² = Q²/(2C) | Two equivalent forms |
Series capacitors | 1/C_eq = Σ(1/Cᵢ) | Always smaller than smallest |
Parallel capacitors | C_eq = ΣCᵢ | Always larger than largest |
Ohm's law | V = IR | |
Parallel resistors | 1/R_eq = Σ(1/Rᵢ) | |
Series resistors | R_eq = ΣRᵢ | |
RC charging | q(t) = Q_max(1 − e^(−t/RC)) | Q_max = εC |
RC discharging | q(t) = Q₀e^(−t/RC) | τ = RC |
RC circuits are everywhere in electronics. The flash on a camera charges a capacitor through a resistor and then discharges it rapidly through the flash tube. Touchscreens detect your finger by measuring changes in capacitance. Defibrillators store energy in a large capacitor and discharge it through the patient's chest, and the RC time constant determines how quickly the energy is delivered.
Students often think that if E = 0 in a region, then V must also be zero. V must be constant (no change), but that constant can be any value, including zero.
A frequent error with dielectrics: students assume inserting a dielectric always increases stored energy. It depends on whether the capacitor is connected to a battery (constant V, energy increases) or isolated (constant Q, energy decreases).
Students mix up the series/parallel formulas for capacitors and resistors. Remember: capacitors in series use the reciprocal sum (like resistors in parallel), and vice versa.
In RC circuits, students sometimes think the capacitor reaches 63% charge at time t = 2RC or t = RC/2. The 63% mark is at exactly one time constant, t = τ = RC.
⚠️ Know the distinction between E = 0 and V = 0. If E = 0 everywhere in a region, V is constant there, but not necessarily zero.
⚠️ Dielectric problems will specify whether the capacitor is connected to a battery or isolated. Read carefully, because the physics is different in each case.
⚠️ Capacitors in series: use the reciprocal formula. A common numerical question is C₁ = 2 μF, C₂ = 3 μF in series → 1.2 μF.
⚠️ Kirchhoff's junction rule is conservation of charge. The loop rule is conservation of energy. Exams ask you to identify which principle each rule represents.
⚠️ RC circuit timing: one time constant = 63% charged (or 37% remaining when discharging). Two time constants for discharging → Q₀e⁻² ≈ 13.5% remaining.
⚠️ The derivation of q(t) for an RC charging circuit from Kirchhoff's loop rule is a classic long-answer question.
True or false: If the electric field is zero in a region, the potential must be zero there.
False. V must be constant, but it can be any constant value.
Fill in the blank: Replacing one of two equal positive charges with a negative charge of the same magnitude changes the system's potential energy from U to ______.
−U
True or false: Inserting a dielectric into a capacitor connected to a battery increases the stored energy.
True. C increases, V stays constant, so U = ½CV² increases.
Fill in the blank: Three 30 Ω resistors in parallel have an equivalent resistance of ______.
10 Ω
Fill in the blank: In an RC circuit, the charging capacitor reaches 63% of its maximum charge after a time of ______.
τ = RC (one time constant)
Q: A system of two positive charges Q₁ and Q₂ is separated by distance d. A negative charge −q is brought from infinity to the midpoint. Is the work done by the electric field positive, negative, or zero?
A: Positive. The negative charge is attracted toward both positive charges, so the electric field does positive work as it pulls the charge in from infinity.
Q: In a region where the electric field is zero, what must be true about the potential V?
A: V must be constant (it can be zero or non-zero). Since E = −dV/dx, zero field means V is not changing.
Q: A parallel-plate capacitor is connected to a battery at voltage V. A dielectric with κ > 1 is inserted. What happens to the stored energy U?
A: U increases. The capacitance increases by factor κ while V is held constant by the battery, so U = ½CV² increases by factor κ.
Q: Two capacitors, C₁ = 2 μF and C₂ = 3 μF, are connected in series. What is the equivalent capacitance?
A: C_eq = (2 × 3)/(2 + 3) = 6/5 = 1.2 μF.
Q: Kirchhoff's junction rule is a statement of which conservation law?
A: Conservation of charge. Current in equals current out at every junction.
Q: A discharging RC circuit has τ = 10 s and initial charge Q₀. What is the charge after 20 s?
A: q = Q₀e^(−20/10) = Q₀e⁻² ≈ 0.135 Q₀.
Q: A wire of length 2.0 m and resistance 10 Ω is connected to a 5 V battery. What is the electric field inside the wire?
A: E = V/L = 5/2.0 = 2.5 V/m.
Q: Derive the expression for charge on a capacitor as a function of time in an RC charging circuit.
A: Start from Kirchhoff's loop rule: ε − IR − q/C = 0. Substitute I = dq/dt to get ε − R(dq/dt) − q/C = 0, which rearranges to dq/(εC − q) = dt/(RC). Integrating with q(0) = 0 gives −ln(εC − q) + ln(εC) = t/(RC), so q(t) = εC(1 − e^(−t/RC)).
Electric potential connects back to Gauss's law (Part 1): knowing E from Gauss's law lets you integrate to find V. It connects forward to magnetism (Part 3), because moving charges (current) create magnetic fields, and circuits provide the steady currents needed to study those fields.
RC circuits also serve as the foundation for AC circuit analysis, which appears in more advanced courses. The exponential charging/discharging behaviour is mathematically identical to many other physical systems (radioactive decay, Newton's law of cooling), so the pattern is worth internalising.
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