A cone made of insulatong material has a total charge Q spread uniformly over its sloping surface. Calculate the work done in bringing a small test chrge q from infiniy to the apex of the cone. The cone has a slope length L.

To determine the work done in bringing a small test charge \( q \) from infinity to the apex of a cone with a total charge \( Q \) uniformly distributed over its sloping surface, we can use principles from electrostatics. The work done in moving a charge in an electric field is related to the electric potential created by the charge distribution. Let's break this down step by step.

Understanding the Electric Field of the Cone

The cone has a total charge \( Q \) distributed uniformly over its sloping surface. To find the electric field at the apex of the cone, we first need to consider the geometry of the cone and how the charge is distributed.

Electric Field Calculation

For a cone with a slant height \( L \) and a uniform surface charge density \( \sigma \), the surface charge density can be expressed as:

• \( \sigma = \frac{Q}{A} \)

where \( A \) is the surface area of the cone. The surface area \( A \) of a cone is given by:

• \( A = \pi r L \)

Here, \( r \) is the radius of the base of the cone. The electric field \( E \) at the apex due to a uniformly charged cone can be derived using integration or symmetry arguments. However, for simplicity, we can use the known result for the electric field at the apex of a cone:

• \( E = \frac{Q}{4 \pi \epsilon_0 L^2} \)

where \( \epsilon_0 \) is the permittivity of free space.

Calculating the Work Done

The work \( W \) done in bringing the test charge \( q \) from infinity to the apex of the cone is given by the formula:

• \( W = q \cdot V \)

Here, \( V \) is the electric potential at the apex of the cone. The electric potential \( V \) due to a charge distribution is related to the electric field by:

• \( V = -\int_{\infty}^{r} E \cdot dr \)

For our case, since we are moving from infinity to the apex, we can express the potential at the apex as:

• \( V = \frac{Q}{4 \pi \epsilon_0 L} \)

Final Work Calculation

Substituting the expression for \( V \) into the work formula gives us:

• \( W = q \cdot \frac{Q}{4 \pi \epsilon_0 L} \)

Thus, the total work done in bringing the small test charge \( q \) from infinity to the apex of the cone is:

• W = \frac{qQ}{4 \pi \epsilon_0 L}

This result shows that the work done depends on the magnitudes of the charges involved, the distance \( L \), and the fundamental constant \( \epsilon_0 \). It illustrates how electric fields and potentials work together to influence the movement of charges in electrostatic scenarios.