GATE 2022 PH – Question 65
A current of 1 A is flowing through a very long solenoid made of winding density 3000 turns/m. As shown in the figure, a parallel plate capacitor, with plates oriented parallel to the solenoid axis and carrying surface charge density 6ε C m2, is placed at the middle of the solenoid. The momentum density of the electromagnetic field at the midpoint X of the capacitor is n× 10ିଵଷ N s m3. The value of n is ______________ (Round off to the nearest integer) (speed of light c= 3 × 10଼ m s1)
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Correct answer: 2
Explanation
The electromagnetic momentum density in crossed electric and magnetic fields is
$$\mathbf g=\epsilon_0\,\mathbf E\times\mathbf B .$$
**Electric field between the capacitor plates:** $E=\dfrac{\sigma}{\epsilon_0}=\dfrac{6\epsilon_0}{\epsilon_0}=6$ V/m.
**Magnetic field inside the solenoid:** $B=\mu_0nI=\mu_0\times3000\times1$.
The plates are parallel to the axis, so $\mathbf E$ is perpendicular to the solenoid axis (and to $\mathbf B$), and
$$g=\epsilon_0EB=\epsilon_0\times6\times\mu_0\times3000=\frac{6\times3000}{c^2}\quad\left(\epsilon_0\mu_0=\frac1{c^2}\right).$$
$$g=\frac{18\,000}{9\times10^{16}}=2\times10^{-13}\ \text{N s m}^{-3}\;\Rightarrow\;n=\mathbf{2}.$$