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Two thin conducting half plates at \phi=0 and \phi=\frac{\pi}{6}, are insulated from each other along the z-axes. Given that the potential function for 0 \leq \phi \leq \frac{\pi}{6} is V=\left(-\frac{60 \phi}{\pi}\right) V, the energy stored between the half planes for 0.1 \leq r \leq 0.6 \mathrm{~m} and 0 \leqslant z \leqslant 1 \mathrm{~m} (Assume free space)

Option: 1

1nJ


Option: 2

1.5nJ


Option: 3

2.5nJ


Option: 4

3.5 nJ


Answers (1)

best_answer

\begin{aligned} E=-\nabla \phi & =-\frac{1}{\partial} \frac{\partial}{\partial \phi}\left(-\frac{60 \phi}{\pi}\right) a_\phi \\ & =\frac{60}{\pi r} a_\phi(V / m) \end{aligned}
\begin{aligned} \omega_E= & \frac{\epsilon_0}{2} \int E^2 d r \\ & =\frac{\epsilon_0}{2} \int_0^1 \cdot \int_0^{\pi / 6} \int_{0.1}^{0.6}\left(\frac{60}{\pi r}\right)^2 r\ d r\ d z \\ & =\frac{300 \epsilon_0}{\pi} \ln 6=1.51 \mathrm{hJ} \text { } \end{aligned}

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Gautam harsolia

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