Quick answer: Electric flux is Φ = ∮E · dA, the field passing through a surface. This AP Physics C lab on gauss's law gives the aim, the equipment, the safety points, a step-by-step method, sample results and 7 analysis questions; the answers are on a separate page and the lab sheet is a free printable PDF.
A charged ball hangs between two parallel plates and swings away from vertical when the voltage is turned on. How does the sideways force depend on the field, and what charge does the ball carry? This lab measures the deflection against voltage and links it to Gauss’s law for the plates.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 8
Aim
To show that the force on a charged ball between parallel plates is proportional to the field E = V/d, and to estimate the charge on the ball and the surface charge density on the plates.
Equipment
- Two parallel metal plates 0.080 m apart on insulating stands
- A variable high-voltage supply (0 to 5 kV) with a protective 5 MΩ resistor
- A foil-coated polystyrene ball of mass 0.12 g on a thin nylon thread 0.600 m long
- A ruler and a plain background card
Safety
Use only a supply designed for school use with a built-in 5 MΩ safety resistor. Switch off and discharge the plates with an insulated lead before touching anything. Teacher supervision is required. Keep away from anyone with a pacemaker.
Method
- Charge the ball once by touching it to the positive terminal, then hang it midway between the plates with no voltage applied.
- Set the supply to 1.0 kV and measure the sideways displacement x of the ball with the ruler.
- Increase V in steps of 1.0 kV up to 5.0 kV, measuring x each time.
- Calculate E = V/d and tanθ = x / √(L² − x²) with L = 0.600 m.
- Plot tanθ against E and find the gradient.
Results
Sample measurements:
| V (kV) | E = V/d (10⁴ N/C) | x (mm) | tanθ |
|---|---|---|---|
| 1.0 | 1.25 | 25 | 0.042 |
| 2.0 | 2.50 | 51 | 0.085 |
| 3.0 | 3.75 | 76 | 0.128 |
| 4.0 | 5.00 | 100 | 0.169 |
| 5.0 | 6.25 | 125 | 0.213 |
Analysis questions
Question 1 Check the table
Verify the value of tanθ for V = 4.0 kV.Question 2 Why E = V/d
Explain why the field between large parallel plates is uniform and equal to σ/ε₀, and why this gives V/d.Question 3 Equilibrium condition
Show that tanθ = qE/(mg) for the ball in equilibrium.Question 4 Charge on the ball
Find the gradient of the tanθ–E graph using the first and last points, and calculate the charge on the ball.Question 5 Surface charge density
Calculate the surface charge density on the plates at 5.0 kV.Question 6 Uncertainty
The displacement is read to ±2 mm. Calculate the percentage uncertainty in x at 5.0 kV and in the charge.Question 7 Improving the experiment
Suggest two sources of error and two improvements.
