Question 103: Electrostats & Capacitance

Question

A spherical metal ball of density '$$\rho$$' and radius '$$r$$' is immersed in a liquid of density '$$\sigma$$'. When an electric field is applied in the upward direction the metal ball remains just suspended in the liquid. Then the expression for the charge on the metal ball is :

Accepted Answer

To find the expression for the charge on the metal ball, we need to consider the forces acting on the ball when it is just suspended in the liquid due to the electric field. The forces involved are:
The weight of the ball (gravitational force), $$F_{	ext{gravity}} = mg$$, where $$m = 	ext{mass of the ball} = ho V = ho \left(\frac{4}{3}\pi r^3ight)$$, and $$g$$ is the acceleration due to gravity.
The buoyant force due to the liquid, which can be given by Archimedes' principle, $$F_{	ext{buoyant}} = 	ext{Volume of displaced liquid} 	imes g 	imes 	ext{density of liquid} = V\sigma g = \left(\frac{4}{3}\pi r^3ight)\sigma g$$.
The electric force in the upward direction which keeps the ball suspended, $$F_{	ext{electric}} = qE$$, where $$E$$ is the electric field strength.
For the ball to be just suspended, the net force acting on it must be zero. This implies:
$$F_{	ext{electric}} = F_{	ext{gravity}} - F_{	ext{buoyant}}$$
Substituting the expressions we have for each:
$$qE = ho \left(\frac{4}{3}\pi r^3ight) g - \sigma \left(\frac{4}{3}\pi r^3ight) g$$
Isolating $$q$$, we find:
$$q = \frac{\left(ho \left(\frac{4}{3}\pi r^3ight) g - \sigma \left(\frac{4}{3}\pi r^3ight) g ight)}{E}$$
Simplifying the expression:
$$q = \frac{\left(\frac{4}{3}\pi r^3ight) (ho - \sigma) g}{E}$$
So, the correct expression for the charge on the metal ball, as per the given options, is option B:
$$q=\frac{\left[4 \pi r^3(ho-\sigma) gight]}{3 E}$$

Answer

$$q=\frac{\left[\frac{4}{3} \pi r^3 \rho g\right]}{\sigma E}$$

Answer

$$q=\frac{\left[4 \pi r^3(\rho-\sigma) g\right]}{3 E}$$

Answer

$$q=\frac{\left[4 \pi r^2 \rho g\right]}{\sigma E}$$

Answer

$$q=\frac{\left[4 \pi r^2(\rho-\sigma) g\right]}{E}$$

Electrostats & Capacitance

A spherical metal ball of density '$\rho' and radius 'r' is immersed in a liquid of density '\sigma$'. When an electric field is applied in the upward direction the metal ball remains just suspended in the liquid. Then the expression for the charge on the metal ball is :

Choose the correct answer: