Two charged infinite thin plane sheets A and B having charge densities \( 3\sigma \) and \( -\sigma \), respectively, are placed parallel and close to each other horizontally with B above A. A particle having positive charge ‘\( q \)’ and mass ‘\( m \)’ is placed between these sheets and released from rest under gravity. The acceleration of the particle is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{2q\sigma}{m\varepsilon_{0}}-g \)
An infinite plane sheet of charge density \( \sigma \) produces a uniform field \( \sigma/2\varepsilon_0 \), directed away from a positive sheet and towards a negative sheet. Between the sheets the two fields are added with their directions.
Taking upwards as positive (B is above A):
Sheet A (\( +3\sigma \), below the particle) pushes the field away from itself, i.e. upwards:
\( \displaystyle E_A = +\frac{3\sigma}{2\varepsilon_0} \)
Sheet B (\( -\sigma \), above the particle) pulls the field towards itself, i.e. also upwards:
\( \displaystyle E_B = +\frac{\sigma}{2\varepsilon_0} \)
\( \displaystyle E = E_A+E_B = \frac{3\sigma}{2\varepsilon_0}+\frac{\sigma}{2\varepsilon_0} = \frac{2\sigma}{\varepsilon_0}\ \text{(upwards)} \)
The electric force on +q is upwards and gravity is downwards, so by Newton’s second law:
\( \displaystyle ma = qE-mg \Rightarrow a = \frac{2q\sigma}{m\varepsilon_0}-g \)