A small charged sphere of mass 2.0 \times 10^{-4}\text{ kg} carrying a positive charge of … — JAMB UTME

A small charged sphere of mass $2.0 \times 10^{-4}\text{ kg}$ carrying a positive charge of $+4.0 \times 10^{-8}\text{ C}$ is suspended by a light insulating string between two vertical parallel plates. When a uniform horizontal electric field of magnitude $E$ is applied between the plates, the string deflects and comes to equilibrium at an angle of $45^\circ$ to the vertical. Taking the acceleration due to gravity $g = 10\text{ m}\cdot\text{s}^{-2}$ , calculate the magnitude of the electric field intensity $E$ in $\text{N}\cdot\text{C}^{-1}$ .

Answer: 50000 N/C

Answer

The magnitude of the electric field intensity is

50000\text{ N}\cdot\text{C}^{-1}

(or

5.0 \times 10^4\text{ N}\cdot\text{C}^{-1}

In electrostatic equilibrium, the sphere experiences three forces: weight (

mg

) vertically downward, electrostatic force (

qE

) horizontally, and tension (

T

) along the thread at

45^\circ

to the vertical. Balancing components gives

T \sin 45^\circ = qE

and

T \cos 45^\circ = mg

. Dividing these yields

\tan 45^\circ = \frac{qE}{mg} = 1

, which gives

qE = mg

. Substituting the given values gives

E = \frac{2.0 \times 10^{-3}}{4.0 \times 10^{-8}} = 50000\text{ N}\cdot\text{C}^{-1}

Step-by-Step Solution

Calculate the weight of the charged sphere.

W = mg = (2.0 \times 10^{-4}\text{ kg})(10\text{ m}\cdot\text{s}^{-2}) = 2.0 \times 10^{-3}\text{ N}

The weight provides the downward vertical force in equilibrium.

Relate the electrostatic force to the weight using the angle of deflection.

\tan(45^\circ) = \frac{F_e}{W} \implies 1 = \frac{F_e}{2.0 \times 10^{-3}\text{ N}} \implies F_e = 2.0 \times 10^{-3}\text{ N}

In electrostatic equilibrium, the ratio of the horizontal force to the vertical force equals the tangent of the angle with the vertical.

Calculate the electric field strength

E

using

F_e = qE

E = \frac{F_e}{q} = \frac{2.0 \times 10^{-3}\text{ N}}{4.0 \times 10^{-8}\text{ C}} = 50000\text{ N}\cdot\text{C}^{-1}

The electric field intensity is the electric force per unit charge.

Key Concept

Equilibrium of a charged body in a uniform electric field

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic