Forces on Charged Pendulum Bob
At equilibrium, the pendulum bob experiences three forces:
- Gravitational Force ($F_g$): acting vertically downwards, $F_g = mg$.
- Electrostatic Force ($F_e$): acting horizontally due to the electric field of the charged sheet.
- Tension ($T$): acting along the string towards the point of suspension.
Electric Field and Electrostatic Force
The electric field ($E$) produced by a vertical sheet of charge with surface charge density $\sigma$ at a distance from the sheet is given by:
$E = \frac{\sigma}{2 \varepsilon_0}$
This electric field exerts a horizontal electrostatic force ($F_e$) on the charged bob ($q$):
$F_e = qE = q \left( \frac{\sigma}{2 \varepsilon_0} \right) = \frac{\sigma q}{2 \varepsilon_0}$
Equilibrium Condition
When the string makes an angle $\theta$ with the vertical, the forces are balanced. Resolving the tension force into horizontal and vertical components:
- Vertical component: $T \cos\theta = mg$
- Horizontal component: $T \sin\theta = F_e = \frac{\sigma q}{2 \varepsilon_0}$
Dividing the horizontal component equation by the vertical component equation:
$\frac{T \sin\theta}{T \cos\theta} = \frac{\frac{\sigma q}{2 \varepsilon_0}}{mg}$
$\tan\theta = \frac{\sigma q}{2 \varepsilon_0 m g}$
Result
The equilibrium condition is met when $\tan\theta = \frac{\sigma q}{2 \varepsilon_0 m g}$.
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