Problem 157 - Master Resource Book in JEE Main Physics

Question

Which of the following statement(s) is/are correct? [IIT JEE 2011] (a) If the electric field due to a point charge varies as \(r^{-2 s}\) instead of \(r^{-2}\), then the Gauss' law will still be valid (b) The Gauss' law can be used to calculate the field distribution around an electric dipole (c) If the electric field between two point charges is zero somewhere, then the sign of the two charges is the same (d) The work done by the extemal force in moving a unit positive charge from point \(A\) at potential \(V_{A}\) to point \(B\) at potential \(V_{B}\) is \(\left(V_{B}-V_{A}\right)\)

Step-by-Step Solution

Answer

Correct statements: (a) and (d).

Step 1: Understanding Gauss' Law

Gauss' Law states that the electric flux through a closed surface is proportional to the charge enclosed. It is mathematically expressed as \( \oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0} \). It is based on the inverse-square law, but it applies to any type of radial field if the field can be enclosed in a symmetric surface.

Step 2: Analyzing Statement (a)

If the electric field due to a point charge varies as \(r^{-2s}\), it still represents a radial field that can be accommodated within Gauss's law, provided the symmetry is maintained. Hence, Gauss's law remains valid for any spherical symmetric charges regardless of the power of \(r\) as long as the field is still inverse radial. Thus, statement (a) is correct.

Step 3: Evaluating Statement (b)

Gauss's law is most effectively used with symmetric charge distributions. An electric dipole does not have a symmetric charge distribution that permits the use of Gaussian surfaces to easily compute the field. Thus, statement (b) is incorrect.

Step 4: Checking Statement (c)

The electric field between two charges can be zero at a point not along the line joining them if the charges are of opposite signs. The zero potential point lies closer to the charge of smaller magnitude in case of opposite charges. Therefore, statement (c) is incorrect.

Step 5: Verifying Statement (d)

The work done \(W\) in moving a unit positive charge from point \(A\) to \(B\) is given by the potential difference \(V_B - V_A\). Therefore, this statement is derived directly from the definition of potential difference. Thus, statement (d) is correct.

Key Concepts

Electric Field

The electric field is a fundamental concept in understanding electric forces. It is a vector field around a charged object where the magnitude and direction of the force can be measured. Electric fields emanate outward from positive charges and inward towards negative charges.
An electric field due to a point charge is typically described by the formula:

• \( E = k \cdot \frac{q}{r^2} \)

Here, \(E\) is the electric field strength, \(k\) is Coulomb's constant, \(q\) is the charge, and \(r\) is the distance from the charge. This shows an inverse-square relationship, meaning the field decreases with the square of the distance.
Understanding this helps in comprehending more complex phenomena, such as the effects of electric fields from multiple charges or different kinds of charge distributions.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. It is represented as a pair of charge \(+q\) and \(-q\) separated by a distance \(d\).
The electric field of a dipole isn't symmetrical—this is critical in understanding why Gauss's Law isn't effective in calculating the field distribution around a dipole. Gauss's Law is best applied to highly symmetrical systems such as spherical or cylindrical charge distributions, where the symmetry simplifies the calculations.
However, electric dipoles are crucial in many applications, and understanding their field distribution requires considering their unique vector nature rather than relying on symmetry alone.

Potential Difference

Potential difference, also known as voltage, is the difference in electric potential between two points in an electric field. It is a measure of the work needed to move a unit positive charge from one point to the other.
The potential difference \( V_B - V_A \) between two points \( A \) and \( B \) relates directly to the work done by an external force when moving a charge between those points.
The work \(W\) done is given by:

• \( W = V_B - V_A \)

This formulation is key to solving many problems in electrostatics, especially in finding the energy required to move charges in electric fields. Understanding potential difference assists in tackling practical and theoretical physics problems effectively.