Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres. The spheres are suspended by non-conducting thread. Figure 3 Each sphere has a mass of 3.2 × 10⁻³ kg and a radius of 20 mm. The distance d is 40 mm. The capacitance of a sphere of radius r is 4πε₀r. Each sphere is charged by connecting it briefly to the positive terminal of a high-voltage supply, the other terminal of which is at 0 V. After this has been done the charge on each sphere is 52 nC. Calculate the potential of one of the spheres. The charged spheres in Figure 3 are at equilibrium. Draw labelled arrows on Figure 3 to show the forces on sphere B. Suggest a solution to one problem involved in the measurement of d in Figure 3. Show that the magnitude of the electrostatic force on each sphere is about 4 × 10⁻³ N. A student measures the angle θ when the apparatus in Figure 3 is at equilibrium. The student records θ as 7°. Discuss whether this measurement is consistent with the other data in this investigation. The student says that the gravitational force between the two spheres has no significant effect on the angle at which the spheres are in equilibrium. Deduce with a calculation whether this statement is valid.

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Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Question 4

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Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres. The spheres are suspended by non-conducting ... show full transcript

Worked Solution & Example Answer:Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Step 1

Calculate the potential of one of the spheres.

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Answer

To calculate the potential (V) of one sphere, we use the formula:

$V = \frac{Q}{C}$

Where:

Q = charge of the sphere = 52 nC = 52 × 10⁻⁹ C
C = capacitance of the sphere = 4πε₀r

The radius r = 20 mm = 20 × 10⁻³ m, and ε₀ = 8.85 × 10⁻¹² F/m:

$C = 4\pi(8.85 \times 10^{-12})(20 \times 10^{-3})$

Calculating C gives:

$C ≈ 2.22 \times 10^{-10} F$

Now substituting back to find V:

$V = \frac{52 \times 10^{-9}}{2.22 \times 10^{-10}} ≈ 234.2 V$

Thus, the potential of one of the spheres is approximately 234 V.

Step 2

Draw labelled arrows on Figure 3 to show the forces on sphere B.

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Answer

For sphere B, the forces need to be represented by arrows:

The electrostatic repulsion force ( $F_{e}$ ) acting horizontally away from sphere A, labeled as 'Electrostatic Force'.
The weight force ( $F_{g}$ ) acting vertically downward due to gravity, labeled as 'Weight'.
The tension in the supporting thread ( $T$ ) acting at an angle towards the support, labeled as 'Tension'.

The arrows should clearly indicate directions with respect to sphere B.

Step 3

Suggest a solution to one problem involved in the measurement of d in Figure 3.

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One problem in measuring the distance d is that the spheres may swing slightly due to air currents or external vibrations. A suggested solution is to use a measuring device that is fixed and does not disturb the setup, such as a digital caliper, to measure the distance between the spheres while ensuring minimal movement.

Step 4

Show that the magnitude of the electrostatic force on each sphere is about 4 × 10⁻³ N.

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Answer

Using Coulomb's law, the electrostatic force (F) between two charges is given by:

$F = k \frac{Q_1 Q_2}{r^2}$

Where:

$k ≈ 8.99 × 10⁹ N m²/C²$ (Coulomb's constant)
$Q_1 = Q_2 = 52 × 10^{-9} C$ (charges on the spheres)
$r = d = 40 × 10^{-3} m$

Substituting the values:

$F = (8.99 × 10^9) \frac{(52 × 10^{-9})^2}{(40 × 10^{-3})^2} ≈ 4.028 × 10^{-3} N$

Thus, the magnitude of the electrostatic force on each sphere is approximately 4 × 10⁻³ N.

Step 5

Discuss whether this measurement is consistent with the other data in this investigation.

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Answer

The measurement of the angle θ can provide insight into the forces acting on the spheres. Since θ = 7°, we can analyze the balance of forces. The sine of the angle can be correlated to the ratio of forces acting on the spheres:

$tan(θ) = \frac{F_{e}}{F_{g}}$

If the ratio aligns with our previously determined forces, this indicates consistency between the angle measurement and the electrostatic force calculated. Thus, measuring the angle is valid in assessing the equilibrium state.

Step 6

Deduce with a calculation whether this statement is valid.

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Answer

To test the statement regarding the gravitational force, we calculate:

$F_{g} = m imes g$

Where:

m = 3.2 × 10⁻³ kg (mass of the sphere)
g = 9.81 m/s² (acceleration due to gravity)

Thus,

$F_{g} = 3.2 × 10^{-3} kg imes 9.81 m/s² ≈ 0.0314 N$

Comparing the gravitational force to the electrostatic force ( $F_{e} ≈ 4 × 10^{-3} N$ ), we find:

$\frac{F_{g}}{F_{e}} ≈ \frac{0.0314}{0.004} = 7.85$

This shows that the gravitational force is significant compared to the electrostatic force, suggesting the student's assertion is not valid.

Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Worked Solution & Example Answer:Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Calculate the potential of one of the spheres.

Draw labelled arrows on Figure 3 to show the forces on sphere B.

Suggest a solution to one problem involved in the measurement of d in Figure 3.

Show that the magnitude of the electrostatic force on each sphere is about 4 × 10⁻³ N.

Discuss whether this measurement is consistent with the other data in this investigation.

Deduce with a calculation whether this statement is valid.

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