Question

In which situation is the electric potential the highest? a) at a point \(1 \mathrm{~m}\) from a point charge of \(1 \mathrm{C}\) b) at a point \(1 \mathrm{~m}\) from the center of a uniformly charged spherical shell with a radius of \(0.5 \mathrm{~m}\) and a total charge of \(1 \mathrm{C}\) c) at a point \(1 \mathrm{~m}\) from the center of a uniformly charged rod with a length of \(1 \mathrm{~m}\) and a total charge of \(1 \mathrm{C}\) d) at a point \(2 \mathrm{~m}\) from a point charge of \(2 \mathrm{C}\) e) at a point \(0.5 \mathrm{~m}\) from a point charge of \(0.5 \mathrm{C}\)

Short answer

Answer: The highest electric potential is \(8.9875x10^9 \mathrm{V}\), found in cases (a) point charge, (b) uniformly charged spherical shell, (d) point charge at a different distance, and (e) point charge with a different charge.

Step-by-step solution

Case (a) - Electric potential due to Point Charge

For a point charge, the electric potential V at a distance r can be calculated using the formula: V = kQ/r where k is the electrostatic constant (\(\approx 8.9875x10^9 \mathrm{Nm^2/C^2}\)), Q is the charge and r is the distance from the charge. In this case, Q = \(1 \mathrm{C}\) and r=\(1 \mathrm{~m}\). So, V1 = k * Q / r = \(8.9875x10^9 \mathrm{V}\).

Case (b) - Electric potential due to Uniformly Charged Spherical Shell

For a uniformly charged spherical shell, the electric potential at any point outside the sphere can be calculated as if the total charge is concentrated at its center. In this case, Q = \(1 \mathrm{C}\), r = \(1 \mathrm{~m}\) (same as previous case). So, V2 = k * Q / r = \(8.9875x10^9 \mathrm{V}\).

Case (c) - Electric potential due to Uniformly Charged Rod

For a uniformly charged rod, the electric potential V at a distance r from its center can be calculated by integrating the point charge electric potential contributions for all the points on the rod. As it is a complex calculation, we can deduce that the electric potential due to the rod will be less than the potential due to a point charge with the same charge, simply because the charge is spread over a larger area. Therefore, V3 < V1.

Case (d) - Electric potential due to Point Charge at different distances

In this case, Q = \(2 \mathrm C\), and r = \(2 \mathrm{~m}\). So, V = k * Q / r = \(8.9875x10^9 \times 2 \mathrm C / 2 \mathrm m=8.9875x10^9 \mathrm{V}\). This implies V4 = V1.

Case (e) - Electric potential due to Point Charge with different charges

In this case, Q = \(0.5 \mathrm C\), and r = \(0.5 \mathrm{~m}\). So, V = k * Q / r = \(8.9875x10^9 \times 0.5 \mathrm C / 0.5 \mathrm m=8.9875x10^9 \mathrm{V}\). This implies V5 = V1.

Comparing the electric potentials

Now we need to compare the electric potentials obtained in all the cases: (a) V1 = \(8.9875x10^9 \mathrm{V}\), (b) V2 = \(8.9875x10^9 \mathrm{V}\), (c) V3 < V1, (d) V4 = \(8.9875x10^9 \mathrm{V}\), and (e) V5 = \(8.9875x10^9 \mathrm{V}\). From the comparison, we can see that cases (a), (b), (d), and (e) all have the highest electric potential, which is equal to \(8.9875x10^9 \mathrm{V}\).