Electric Fields & Capacitance
An electric field is a region of space in which an electrically charged particle experiences an electrostatic force. This guide covers Coulomb's inverse-square law for point charges, uniform electric fields between parallel plates, electric potential, and how capacitors store charge and energy.
- Electric Field Strength (E)
- The force per unit positive charge acting on a small test charge placed in the field (E = F/Q).
- Electric Potential (V)
- The work done per unit positive charge to move a small test charge from infinity to that point in the field.
- Capacitance (C)
- The charge stored per unit potential difference across a capacitor (C = Q/V), measured in Farads.
Electric Field Simulator & Interactive NPN Transistor Builder
Adjust charges and distances to visualize Coulomb’s Law field lines, or switch to the Transistor Lab to wire and test an NPN circuit.
1. Coulomb’s Law Point-Charge Calculation
Formula: F = k · |Q₁ · Q₂| / r² where k = 1/(4πε₀) = 8.99 × 10⁹ N·m²·C⁻²
3.60 N
Attractive (towards each other)7.19e+6 N/C
At r/2 = 0.10 m2. Uniform Parallel-Plate Field Calculator
Formula: E = V / d (units: V m⁻¹ or N C⁻¹)
Exam notes: Radial vs Uniform Fields, Potential & Motion of Charged Particles
- An electric field is any region of space where a charged particle experiences an electrostatic force. Field lines always point away from positive (+Q) and towards negative (−Q), showing the direction of force on a positive test charge.
- In a radial field around a point charge Q, field strength follows an inverse-square law: E = Q / (4πε₀r²). Doubling the distance r reduces E by a factor of 4, whereas electric potential V = Q / (4πε₀r) decreases inversely with r (1/r).
- Between parallel plates, the electric field is uniform (parallel, equally spaced lines): E = V / d. A charged particle (+q, mass m) entering perpendicular to a uniform electric field experiences a constant vertical acceleration a = Eq / m and follows a parabolic trajectory (just like a projectile in uniform gravity).
- Equipotential surfaces are always perpendicular (90°) to electric field lines. Moving a charge along an equipotential requires zero work (ΔW = qΔV = 0 because ΔV = 0).
| Feature | Electric Field (Radial) | Gravitational Field (Radial) |
|---|---|---|
| Origin of field | Charge Q (Coulombs, C) | Mass M (Kilograms, kg) |
| Force law | Coulomb's Law: F = Q₁Q₂ / (4πε₀r²) | Newton's Law: F = −GMm / r² |
| Direction of force | Attractive (opposite signs) OR Repulsive (like signs) | Always attractive only |
| Field strength definition | Force per unit positive charge: E = F / q (N C⁻¹ or V m⁻¹) | Force per unit mass: g = F / m (N kg⁻¹) |
| Potential formula | V = Q / (4πε₀r) (positive or negative) | V_g = −GM / r (always negative; zero at infinity) |
| Shielding possible? | Yes (a Faraday cage or conductor shields electric fields) | No (gravitational fields cannot be shielded) |
Key Concepts & Exam Definitions (10)
- Electric Field Strength (E)
- The force per unit positive charge acting on a small test charge placed in the field (E = F/Q).
- Electric Potential (V)
- The work done per unit positive charge to move a small test charge from infinity to that point in the field.
- Capacitance (C)
- The charge stored per unit potential difference across a capacitor (C = Q/V), measured in Farads.
- Time Constant (τ)
- The time taken for a discharging capacitor's charge to fall to 1/e (about 37%) of its initial value, equal to R × C.
- Dielectric
- An insulating material placed between the plates of a capacitor that increases its capacitance by polarizing in the electric field.
- Electric Field Strength (E) as Potential Gradient
- E = −ΔV / Δr (or E = V / d in a uniform field). The negative sign shows that electric potential V decreases in the direction of the electric field lines. Units: V m⁻¹ is identical to N C⁻¹.
- Work Done Moving a Charge (ΔW = QΔV)
- The work done moving a charge Q across a potential difference ΔV is ΔW = QΔV. One electronvolt (1 eV = 1.60 × 10⁻¹⁹ J) is the kinetic energy gained by an electron accelerated through a potential difference of 1 V.
- Parabolic Motion in a Uniform Field
- When an electron enters horizontally at speed v_x between charged parallel plates, horizontal velocity stays constant (v_x = constant), while vertical acceleration is constant: a_y = eE / m = eV / (md), producing a parabolic path.
- Millikan's Oil Drop Condition (Stationary Charged Drop)
- A charged oil droplet of mass m and charge q held stationary between horizontal parallel plates has upward electric force equal to downward weight: qE = mg, or q(V/d) = mg.
- Graph of E vs r and V vs r for a Conducting Sphere
- Inside a charged hollow conducting sphere (r < R), the electric field E = 0 everywhere and the electric potential V is constant (equal to the surface value V = Q / 4πε₀R). Outside (r ≥ R), E ∝ 1/r² and V ∝ 1/r.
Quick-Fire Flashcards (10)
Click any card to check the answer, then mark it as Known to track your progress.
How is electric field strength related to potential difference in a uniform field?
What does the area under a charge-potential difference (Q-V) graph represent?
How does an electric field line point?
Is electric potential a scalar or vector quantity?
What is the equation for exponential decay of charge on a capacitor?
What is the electric field strength E and electric potential V inside a hollow charged metal sphere of radius R?
Why does a charged particle move in a parabola when fired perpendicular to a uniform electric field, but in a circle in a uniform magnetic field?
State the equation relating electric field strength E to the gradient of an electric potential against distance (V-r) graph.
How much work is done moving a +5.0 µC charge along an equipotential line of +200 V?
An oil drop of mass m and charge q is held stationary between two horizontal plates separated by distance d at voltage V. Write the equilibrium equation.
Exam-Style Questions & Mark Schemes (7)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
Explain how a dielectric increases the capacitance of a capacitor.
Click “Check Mark Scheme” to compare your answer, A dielectric contains polar molecules. When an electric field is applied between the plates, these molecules align with their positive ends toward the negative plate and vice versa. This polarization creates an opposing electric field, which reduces the overall electric field between the plates. Since E = V/d, the potential difference V decreases. Because C = Q/V, for a given charge Q, a lower V means the capacitance C is increased.
- Q2
Compare and contrast electric and gravitational fields.
Click “Check Mark Scheme” to compare your answer, Similarities: Both obey an inverse-square law for force and field strength. Both have field lines and equipotential surfaces. Both have potential inversely proportional to distance. Differences: Gravitational forces are only attractive, while electric forces can be attractive or repulsive. Gravitational fields act on mass, while electric fields act on charge.
- Q3
A 100 μF capacitor is charged to 12 V and discharged through a 50 kΩ resistor. Calculate the time constant and the charge remaining after 5 seconds.
Click “Check Mark Scheme” to compare your answer, Time constant τ = RC = (50 x 10^3) * (100 x 10^-6) = 5 seconds. Initial charge Q0 = CV = (100 x 10^-6) * 12 = 1.2 x 10^-3 C. Using Q = Q0 e^(-t/RC): Q = (1.2 x 10^-3) * e^(-5/5) = (1.2 x 10^-3) * e^-1 = 4.4 x 10^-4 C.
- Q4
Two point charges of +4.0 µC and −9.0 µC are placed 0.30 m apart in a vacuum. Calculate the magnitude of the electrostatic force between them. [2 marks]
Click “Check Mark Scheme” to compare your answer, Using Coulomb's law: F = Q₁Q₂ / (4πε₀r²) = (8.99 × 10⁹ × 4.0 × 10⁻⁶ × 9.0 × 10⁻⁶) / (0.30)² = 0.3236 / 0.090 = 3.6 N (attractive force). [2 marks]
- Q5
An electron (q = 1.60 × 10⁻¹⁹ C, m = 9.11 × 10⁻³¹ kg) enters horizontally at 2.0 × 10⁷ m/s midway between two parallel plates 2.0 cm apart with a potential difference of 400 V. Calculate its vertical acceleration between the plates. [3 marks]
Click “Check Mark Scheme” to compare your answer, 1) Uniform field strength E = V / d = 400 V / 0.020 m = 2.0 × 10⁴ V m⁻¹ [1 mark]. 2) Electric force F = eE = 1.60 × 10⁻¹⁹ × 2.0 × 10⁴ = 3.2 × 10⁻¹⁵ N [1 mark]. 3) Acceleration a = F / m = 3.2 × 10⁻¹⁵ / 9.11 × 10⁻³¹ = 3.51 × 10¹⁵ m/s² towards the positive plate [1 mark].
- Q6
Compare and contrast the electric field around a proton with the gravitational field around a proton. [4 marks]
Click “Check Mark Scheme” to compare your answer, Similarities: Both are radial fields; both obey an inverse-square law with distance (F ∝ 1/r² and field strength ∝ 1/r²); both have field potentials inversely proportional to r (V ∝ 1/r) [2 marks]. Differences: The gravitational field is always attractive (acts on mass), whereas the electric field acts on charge and can be attractive (with electrons) or repulsive (with other protons); the electrostatic force is roughly 10³⁶ times stronger than the gravitational force at the same separation [2 marks].
- Q7
Explain why no work is done when a point charge is moved along an equipotential surface around an isolated positive charge. [2 marks]
Click “Check Mark Scheme” to compare your answer, An equipotential surface is a line/surface of constant electric potential, so the potential difference between any two points on it is ΔV = 0 [1 mark]. Since work done W = QΔV (or because the electric force is always perpendicular to the direction of motion along the equipotential), W = 0 J [1 mark].
Common Exam Questions (3)
What is an electric field?
An electric field is a region surrounding a charged object where another charge experiences an electrostatic force. Electric field strength E is defined as the force per unit positive test charge: E = F / Q.
What is Coulomb's Law?
Coulomb's Law states that the electrostatic force between two point charges in a vacuum is directly proportional to the product of the charges and inversely proportional to the square of their separation: F = Q₁Q₂ / (4πε₀r²).
What is a capacitor?
A capacitor is an electronic component consisting of two conducting plates separated by an insulating dielectric. When connected to a voltage source, equal and opposite charges (+Q and -Q) build up on the plates, storing electrical energy in the electric field between them.
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