Question

A thin spherical conducting shell of radius \(\mathrm{R}\) has a charge q. Another charge \(Q\) is placed at the centre of the shell. The electrostatic potential at a point p a distance \((\mathrm{R} / 2)\) from the centre of the shell is ..... (A) \(\left[(q+Q) /\left(4 \pi \epsilon_{0}\right)\right](2 / R)\) (B) \(\left[\left\\{(2 Q) /\left(4 \pi \epsilon_{0} R\right)\right\\}-\left\\{(2 Q) /\left(4 \pi \epsilon_{0} R\right)\right]\right.\) (C) \(\left[\left\\{(2 Q) /\left(4 \pi \in_{0} R\right)\right\\}+\left\\{q /\left(4 \pi \epsilon_{0} R\right)\right]\right.\) (D) \(\left[(2 \mathrm{Q}) /\left(4 \pi \epsilon_{0} \mathrm{R}\right)\right]\)

Short answer

The short answer to the problem is: \(V = \frac{2(q+Q)}{4\pi\epsilon_{0}R}\)

Step-by-step solution

Finding potential due to charge q

: To determine the electrostatic potential due to charge q at point p, we need to find the distance between point p and the surface of the shell. Since point p is at a distance R/2 from the center of the shell, and the shell has a radius of R, the distance between point p and the surface of the shell (r) is R/2. Calculate the potential as follows: \(V_{q} = \frac{q}{4\pi\epsilon_{0}(R/2)} = \frac{2q}{4\pi\epsilon_{0}R}\)

Finding potential due to charge Q

: Next, find the electrostatic potential due to charge Q at point p. Since point p is at a distance R/2 from the center of the shell, the distance between charge Q and point p is also R/2. Calculate the potential as follows: \(V_{Q} = \frac{Q}{4\pi\epsilon_{0}(R/2)} = \frac{2Q}{4\pi\epsilon_{0}R}\)

Finding total potential at point p

: Finally, add the potentials due to charges q and Q to find the total potential at point p: \(V = V_{q}+V_{Q} = \frac{2q}{4\pi\epsilon_{0}R} + \frac{2Q}{4\pi\epsilon_{0}R}\) Combine the terms: \(V = \frac{2(q+Q)}{4\pi\epsilon_{0}R}\) Comparing this expression to the given options, we can see that it matches with option (A): \(\left[(q+Q) /\left(4 \pi \epsilon_{0}\right)\right](2 / R)\)

Key concepts

Conducting Shell

A conducting shell is a unique structure in electrostatics. It operates under the principle that free charge resides on its surface. This is due to the repulsive force between like charges, which causes them to spread evenly over the surface. Inside the conducting shell, the electric field is zero. This important concept is known as electrostatic shielding, which means that anything inside the shell is shielded from external electric fields. The charge distribution on a conducting shell depends on external charges as well as its own net charge.

Charge Distribution

In electrostatics, charge distribution determines how charges are spread out in a system. It is essential to understand how charges interact to predict how they will distribute. In our problem, we have two charges: one at the center of the shell and another on the outer shell.

• The charge at the center (charge Q) does not change the net charge on the shell, but it does affect the electric potential due to the influence at inner surfaces.
• The charge on the shell surface (charge q) assumes a distribution that maintains electric equilibrium, meaning it spreads out uniformly on the outer surface of the shell.

The resulting charge distribution causes specific potential differences within or around the shell.

Electrostatics

Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from static electric charges. In solving electrostatic problems, the essential components involve understanding electric forces, electric fields, and electric potentials. Electric potential is a scalar quantity that represents the work done to move a unit charge from a reference point to a specific point in the field. It is important to remember that electric potential within a conductor in electrostatic equilibrium is constant.

• Electric potentials are influenced by the arrangement and magnitude of electric charges.
• In our scenario, potential is evaluated by summing contributions from the center charge and the shell charge.

This understanding guides the effective calculation of potentials in complex charge configurations.

Physics Problem-Solving

Solving physics problems, particularly in electrostatics, involves a step-by-step analytical approach to dissect the problem and apply relevant principles.

• First, identify all charges in the system and their positions.
• Calculate the influences each charge has on the point of interest separately.
• Use superposition to add their effects for the total potential or force.

In our problem, the solution process includes calculating the individual potentials from charges q and Q, resulting from their distances from the point of interest, and then summing these potentials to find the total potential. By breaking the problem into smaller parts, you can manage complex systems efficiently.