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  4. An infinitely long solid cylinder of radius R has a uniform volume charge density r. It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression (23 ρ R/16K ε0) . The value of k is

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Q. An infinitely long solid cylinder of radius R has a uniform volume charge density r. It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression $\frac{23 \rho R}{16K \varepsilon_0} $ . The value of k isPhysics Question Image

Electric Charges and Fields

Solution:

We suppose that the cavity is filled up by a positive as well as negative volume charge of r. So the electric field now produced at P is the superposition of two electric fields.
(i) The electric field created due to the infinitely long solid cylinder is $E_1 = \frac{\rho R}{4 \varepsilon_0}$ directed towards the +Y direction
(ii) The electric field created due to the spherical negative charge density $E_1 = \frac{\rho R}{96 \varepsilon_0}$ directed towards the -Y direction.
$\therefore $ The net electric field is
$E = E_1 - E_2 = \frac{1}{6} \left[ \frac{23 \rho R}{16 \varepsilon_0} \right]$