Question 876: Electrostats & Capacitance

Question

Two spherical conductors of radii $$4 \mathrm{~cm}$$ and $$5 \mathrm{~cm}$$ are charged to the same potential. If '$$\sigma_1$$' and '$$\sigma_2$$' be the respective values of the surface density of charge on the two conductors then the ratio $$\sigma_1: \sigma_2$$ is

Accepted Answer

When two spherical conductors are charged to the same potential, the relationship between the potential ($$V$$), surface charge density ($$\sigma$$), and radius ($$R$$) of the conductor can be utilized to find the ratio of the surface charge densities $$\sigma_1$$ and $$\sigma_2$$. The electric potential on the surface of a charged spherical conductor is given by:
$$V = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{R}$$
Where:
$$V$$ is the potential,
$$Q$$ is the total charge,
$$R$$ is the radius of the sphere,
$$\epsilon_0$$ is the permittivity of free space.
The surface charge density ($$\sigma$$) is defined as the charge per unit area, so:
$$\sigma = \frac{Q}{4\pi R^2}$$
Combining these expressions, we can express the potential in terms of the surface charge density as follows:
$$V = \frac{\sigma R}{\epsilon_0}$$
Given that the two spheres are charged to the same potential ($$V_1 = V_2 = V$$), and the radii are $$R_1 = 4\, \mathrm{cm}$$ and $$R_2 = 5\, \mathrm{cm}$$, we can set up the equality:
$$\frac{\sigma_1 R_1}{\epsilon_0} = \frac{\sigma_2 R_2}{\epsilon_0}$$
Upon simplifying, $$\epsilon_0$$ cancels out on both sides, allowing us to solve for the ratio of $$\sigma_1$$ to $$\sigma_2$$ directly:
$$\sigma_1 R_1 = \sigma_2 R_2$$
Thus,
$$\frac{\sigma_1}{\sigma_2} = \frac{R_2}{R_1}$$
Plugging in the given radii values:
$$\frac{\sigma_1}{\sigma_2} = \frac{5}{4}$$
This corresponds to the ratio of surface charge densities:
$$\sigma_1 : \sigma_2 = 5 : 4$$
Therefore, the correct answer is:
Option A $$5: 4$$

Answer

$$5: 4$$

Answer

$$3: 2$$

Answer

$$4: 3$$

Answer

$$2: 1$$

Electrostats & Capacitance

Two spherical conductors of radii $4 \mathrm{~cm} and 5 \mathrm{~cm} are charged to the same potential. If '\sigma_1' and '\sigma_2' be the respective values of the surface density of charge on the two conductors then the ratio \sigma_1: \sigma_2$ is

Choose the correct answer: