An isolated solid conducting sphere is initially uncharged. Electrons are then transferred to the sphere. State and explain the location of the excess electrons. [2 marks] Figure 3 shows how the electric potential V varies with distance r from the centre of the sphere. The radius of the sphere is 0.10 m. The magnitude of the electric field strength E is related to V by E = -\frac{\Delta V}{\Delta r}. Determine, using this relationship, the magnitude of the electric field strength at a distance 0.30 m from the centre of the sphere. State an appropriate SI unit for your answer. electric field strength = unit = The sphere acts as a capacitor because it stores charge at an electric potential. Show that the capacitance of the sphere is approximately 1 × 10⁻¹¹ F. Electrons leak away from the sphere with time, and the amount of energy stored by the sphere decreases. At one instant, the magnitude of the electric potential of the sphere has fallen to 1.0 × 10⁶ V. Calculate, for this instant, the change in the energy stored by the sphere.

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An isolated solid conducting sphere is initially uncharged - AQA - A-Level Physics - Question 3 - 2022 - Paper 2

Question 3

An isolated solid conducting sphere is initially uncharged. Electrons are then transferred to the sphere. State and explain the location of the excess electrons. [2... show full transcript

Worked Solution & Example Answer:An isolated solid conducting sphere is initially uncharged - AQA - A-Level Physics - Question 3 - 2022 - Paper 2

Step 1

State and explain the location of the excess electrons.

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Answer

The excess electrons are located on the outer surface of the conducting sphere. This is because, in a conductor, excess charge will move to the surface to minimize repulsion between like charges.

Step 2

Determine, using this relationship, the magnitude of the electric field strength at a distance 0.30 m from the centre of the sphere.

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Answer

At a distance of 0.30 m, we can find the electric field strength by determining the gradient from the graph in Figure 3. The change in potential ΔV can be observed from the graph between r = 0.2 m and r = 0.3 m:

At r = 0.2 m, V ≈ -0.8 × 10⁶ V
At r = 0.3 m, V ≈ -1.2 × 10⁶ V

Thus, ΔV = V_final - V_initial = (-1.2 × 10⁶) - (-0.8 × 10⁶) = -0.4 × 10⁶ V.

The change in distance Δr = 0.3 - 0.2 = 0.1 m.

Now, applying the formula:

$E = -\frac{\Delta V}{\Delta r} = -\frac{-0.4 \times 10^6}{0.1} = 4.0 \times 10^6 \text{ N/C}.$

The unit for the electric field strength is N/C.

Step 3

Show that the capacitance of the sphere is approximately 1 × 10⁻¹¹ F.

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Answer

For a sphere, the capacitance can be calculated using the equation:

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

Where Q is the charge and V is the electric potential. To find the charge, we can use the relationship from part 0.3.2. Assuming Q is approximately 2.0 × 10⁻¹⁹ C:

Using the electric potential from the previous part, 1.0 × 10⁶ V:

$C = \frac{2.0 \times 10^{-19}}{1.0 \times 10^6} = 2.0 \times 10^{-25} F.$

However, the capacitance value for a spherical conductor is generally expressed as:

$C = 4\pi\epsilon_0 r$

Where $\epsilon_0 \approx 8.85 \times 10^{-12} F/m$ and r = 0.1 m, resulting in:

$C \approx 4\pi(8.85 \times 10^{-12})0.1 \approx 1 \times 10^{-11} F.$

Step 4

Calculate, for this instant, the change in the energy stored by the sphere.

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Answer

The energy stored in a capacitor is given by the equation:

$E = \frac{1}{2} C V^2$

Using the value of capacitance C ≈ 1 × 10⁻¹¹ F and V = 1.0 × 10⁶ V:

$E = \frac{1}{2} (1 \times 10^{-11}) (1.0 \times 10^6)^2 = \frac{1}{2} (1 \times 10^{-11}) (1.0 \times 10^{12}) = 5.0 \text{ J}.$

Therefore, the change in energy stored by the sphere is 5.0 J.

An isolated solid conducting sphere is initially uncharged - AQA - A-Level Physics - Question 3 - 2022 - Paper 2

Worked Solution & Example Answer:An isolated solid conducting sphere is initially uncharged - AQA - A-Level Physics - Question 3 - 2022 - Paper 2

State and explain the location of the excess electrons.

Determine, using this relationship, the magnitude of the electric field strength at a distance 0.30 m from the centre of the sphere.

Show that the capacitance of the sphere is approximately 1 × 10⁻¹¹ F.

Calculate, for this instant, the change in the energy stored by the sphere.

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