Question

A cone made of insulating material has a total charge Q spread uniformly over its sloping surface.

Calculate the energy required to take a test charge q from infinity to apex A of cone. The slant length is L. कुचालक पदार्थ से बने एक शंकु पर कुल आवेश Q इसकी ढालनुमा सतह पर एकसमान रूप से फैला हुआ है। एक परीक्षण आवेश q को अनन्त से शंकु के शीर्ष A तक लाने में आवश्यक ऊर्जा की गणना कीजिए। ढाल की लम्बाई L है।

Answer 1

\frac{Qq}{2\pi \epsilon_0 L}

Answer 2

1. The energy required to move a test charge $q$ from infinity to point $A$ is $qV_A$, where $V_A$ is the electric potential at point $A$.

2. The potential at $A$ due to the uniformly charged sloping surface is calculated by integrating the potential contributions from infinitesimal charge elements over the surface.

3. A suitable element of charge is a ring of charge on the sloping surface at a slant distance $l'$ from the apex.

4. The potential due to this ring at the apex is calculated using $dV = \frac{1}{4\pi\epsilon_0} \frac{dQ}{l'}$, where $dQ = \sigma dA$ and $dA = 2\pi r' dl'$ is the area of the ring.

5. The radius of the ring $r'$ is related to the slant distance $l'$ by $r' = (R/L)l'$.

6. Integrating $dV$ from $l'=0$ to $l'=L$ gives the total potential $V_A$.

7. Expressing the result in terms of the total charge $Q = \sigma \pi R L$ gives $V_A = \frac{Q}{2\pi L \epsilon_0}$.

8. The required energy is $W = qV_A = \frac{Qq}{2\pi \epsilon_0 L}$.

Answer: $\frac{Qq}{2\pi \epsilon_0 L}$