Question 875: Electrostats & Capacitance

Question

A charged spherical conductor has radius '$$r$$'. The potential difference between its surface and 3 point at a distance '$$3 r$$' from the centre is '$$v$$' The electric intensity at a distance '$$3 r$$' from the centre of the conductor is

Accepted Answer

To determine the electric intensity at a distance $$3r$$ from the center of the charged spherical conductor, we first need to understand the relationship between potential difference and electric field.
The potential due to a charged spherical conductor at a distance $$r$$ from its center is given by:
$$V = \frac{Q}{4 \pi \epsilon_0 r}$$
where $$Q$$ is the charge on the conductor and $$\epsilon_0$$ is the permittivity of free space.
The potential at the surface of the conductor (radius $$r$$) is:
$$V_1 = \frac{Q}{4 \pi \epsilon_0 r}$$
The potential at a distance $$3r$$ from the center is:
$$V_2 = \frac{Q}{4 \pi \epsilon_0 \cdot 3r} = \frac{Q}{12 \pi \epsilon_0 r}$$
The potential difference between the surface and the point at a distance $$3r$$ (given as $$v$$) is:
$$v = V_1 - V_2 = \frac{Q}{4 \pi \epsilon_0 r} - \frac{Q}{12 \pi \epsilon_0 r}$$
Simplifying the difference, we get:
$$v = \frac{Q}{4 \pi \epsilon_0 r} \left(1 - \frac{1}{3}\right) = \frac{Q}{4 \pi \epsilon_0 r} \cdot \frac{2}{3} = \frac{Q}{6 \pi \epsilon_0 r}$$
The electric field intensity at a distance $$3r$$ from the center is given by:
$$E = \frac{Q}{4 \pi \epsilon_0 \cdot (3r)^2} = \frac{Q}{36 \pi \epsilon_0 r^2}$$
From the potential difference equation, we can express $$Q$$ in terms of $$v$$ and substitute it back:
$$Q = \frac{6 \pi \epsilon_0 r v}{1}$$
Thus, the electric field intensity at a distance $$3r$$ is:
$$E = \frac{\frac{6 \pi \epsilon_0 r v}{1}}{36 \pi \epsilon_0 r^2} = \frac{6 v}{36 r} = \frac{v}{6 r}$$
Hence, the electric intensity at a distance $$3r$$ from the center of the conductor is:
Option D: $$\frac{v}{6 r}$$

Answer

$$\frac{v}{8 r}$$

Answer

$$\frac{v}{2 r}$$

Answer

$$\frac{v}{4 r}$$

Answer

$$\frac{v}{6 r}$$

Electrostats & Capacitance

A charged spherical conductor has radius '$r'. The potential difference between its surface and 3 point at a distance '3 r' from the centre is 'v' The electric intensity at a distance '3 r$' from the centre of the conductor is

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