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Chapter 13: Problem 157

A thin spherical conducting shell of radius \(R\) has a charge \(q\). Another charge \(Q\) is placed at the centre of the shell. The electrostatic potential at a point \(P\) at a distance \(R / 2\) from the centre of the shell is [2003] (A) \(\frac{2 Q}{4 \pi \varepsilon_{0} R}\) (B) \(\frac{2 Q}{4 \pi \varepsilon_{0} R}-\frac{2 q}{4 \pi \varepsilon_{0} R}\) (C) \(\frac{2 Q}{4 \pi \varepsilon_{0} R}+\frac{q}{4 \pi \varepsilon_{0} R}\) (D) \(\frac{(q+Q)}{4 \pi \varepsilon_{0}} \frac{2}{R}\)

Short Answer

Expert verified
The short answer to the problem is: The electrostatic potential at point P is given by V_total = \(\frac{2Q}{4\pi \varepsilon_{0}R}+\frac{q}{4\pi \varepsilon_{0}R}\).

Step by step solution

01

Determine the potential at point P due to charge Q

To find the potential at the point P due to charge Q, use the formula V = kQ/r. Here, the charge is Q, distance r is R/2, and k = 1/(4蟺蔚鈧�). Plugging the values, we get: V_Q = \(\frac{Q}{4\pi \varepsilon_{0} (R/2)} = \frac{2Q}{4\pi \varepsilon_{0}R}\)
02

Determine the potential at point P due to charge q

Since the charge q is uniformly distributed across the conducting spherical shell, the potential inside the shell will be constant, making the potential at point P: V_q = \(\frac{q}{4\pi \varepsilon_{0}R}\)
03

Determine the total electrostatic potential at point P

The total electrostatic potential at point P is the sum of the potentials due to charges q and Q. Adding the potentials found in step 1 and step 2, we get: V_total = V_Q + V_q = \(\frac{2Q}{4\pi \varepsilon_{0}R} + \frac{q}{4\pi \varepsilon_{0}R}\) Simplifying the expression, we get: V_total = \(\frac{2Q + q}{4\pi \varepsilon_{0}R}\) Comparing this with the given options, we find it matches with option (C). Therefore, the electrostatic potential at point P is: V_total = \(\frac{2Q}{4\pi \varepsilon_{0}R}+\frac{q}{4\pi \varepsilon_{0}R}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spherical Conducting Shell
A spherical conducting shell is a key concept in electrostatics, often encountered in problems involving electric charges and fields. Imagine a hollow ball made of a conductive material like metal. This shell can carry charge on its surface, and due to its symmetrical shape, the charge distributes itself uniformly over the outer surface. An important property of such conductive shells is that the electric field inside a charged, conducting shell is zero. This results in the electric potential being constant throughout the interior space of the shell, including at its inner surface.

When an external charge is placed inside such a shell, it does not affect the charge distribution on the outside surface. The reverse is also true; charges outside the shell do not influence the charges within. This concept is vital in understanding how to protect sensitive electronic equipment from external electric fields by using a Faraday cage, which is based on the principles of a conducting shell.
Point Charge
The idea of a point charge is a simplification used in physics that refers to an idealized charge of infinitesimal size that is completely characterized by its magnitude. A point charge is a useful concept, as it allows us to calculate electric fields and potentials in various configurations by using Coulomb's law.

In the context of the problem at hand, we consider a point charge Q placed at the center of the spherical conducting shell. The electric potential caused by this point charge at any given radius can be calculated using the formula V = kQ/r, where V is the electric potential, k is Coulomb's constant, Q is the charge, and r is the radial distance from the charge to the point of interest. The simplicity of using point charges helps students to dive into more complex problems involving charge distributions without getting bogged down by complexities at the start.
Electric Potential Due to Charge Distribution
When discussing electric potential due to charge distribution, we're referring to the work needed to move a unit positive charge from a reference point (usually taken at infinity) to a specific point in space, in the presence of a static electric field. This concept is crucial in electrostatics because it helps us to understand and calculate the influence of various charge configurations on nearby points in space.

In our problem, charge distribution consists of a point charge at the center of a spherical shell and the shell itself, which is uniformly charged. The potential at any point inside the shell due to the shell's charge is the same and can be found as if all the charge were concentrated at the center. This is due to the shell鈥檚 spherical symmetry and is backed by Gauss's law. The total electric potential at a point is therefore the sum of potentials due to each charge distribution affecting that point.

It is crucial to add the potentials due to individual charges algebraically, taking care to consider the signs of the charges, to find the total electric potential at a specific point in the field.

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Most popular questions from this chapter

A table tennis ball which has been covered with a conducting paint is suspended by a silk thread so that it hangs between two metal plates. One plate is earthed. When the other plate is connected to a high voltage generator, the ball (A) is attracted to the high voltage plate and stays there. (B) hangs without moving. (C) swings backward and forward hitting each plate in turn. (D) is repelled to the earthed plate and stays there.

A parallel plate capacitor of capacitance \(C\) is connected to a battery of emf \(V .\) If a dielectric slab is completely inserted between the plates of the capacitor and battery remains connected, then electric field between plates (A) decreases. (B) increases. (C) remains constant. (D) may be increase or may be decrease.

A charged particle of mass \(m\) and charge \(q\) is released from rest in an electric field of constant magnitude \(E\). The kinetic energy of the particle after time \(t\) will be (A) \(\frac{2 E^{2} t^{2}}{m q}\) (B) \(\frac{E q^{2} m}{2 t^{2}}\) (C) \(\frac{E^{2} q^{2} t^{2}}{2 m}\) (D) \(\frac{E q m}{2 t}\)

This question has Statement 1 and Statement 2. Of the four choices given after the statements, choose the one that best deseribes the two statements. \(|2012|\) An insulating solid sphere of radius \(R\) has a uniformly positive charge density \(\rho\). As a result of this uniform charge distribution, there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at a point out side the sphere. The electric potential at infinity is zero. Statement 1: When a charge \(q\) is taken from the centre to the surface of the sphere, its potential energy changes by \(\frac{q \rho}{3 \varepsilon_{0}}\) Statement 2: The electric field at a distance \(r(r

Two identical particles each having charge \(q\) are very far apart. They are given velocity \(v_{0}\) parallel to each other such that initial perpendicular separation between them is \(d\). If the subsequent minimum separation between them is \(2 d\), find the initial velocity \(v_{0}\) and the loss in their total kinetic energies.
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