Problem 103 - Master Resource Book in JEE Main Physics

Question

For a point situated outside the sphere at a \(r>R\), the electric field is given by (a) \(E=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{2}}\) (b) \(E=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{-Q}{r^{2}}\) (c) \(E=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{3}}\) (d) \(E=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q^{2}}{r^{2}}\)

Step-by-Step Solution

Answer

The correct option is (a).

Step 1: Understanding the Context

We are given a sphere with a certain charge and we need to determine the electric field at a point outside this sphere where the distance from the center of the sphere to this point is greater than the sphere's radius (i.e., at a distance \(r > R\)). This scenario calls for applying Gauss's Law to find the electric field of a charged sphere from a point outside the surface.

Step 2: Recall the Formula of Electric Field Outside a Charged Sphere

The electric field formula for a point outside a uniformly charged sphere is the same as that for a point charge situated at the center of the sphere. This is given by Gauss’s Law. Hence, the electric field \(E\) at a distance \(r\) from the center of the sphere is:\[ E = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{2}} \]where \(Q\) is the total charge of the sphere.

Step 3: Identify the Correct Option

Examine each of the provided options:(a) \(E = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{2}}\) - This matches the derived formula for an electric field outside a charged sphere.(b) \(E = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{-Q}{r^{2}}\) - This represents a field in the opposite direction for a positive charge.(c) \(E = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{3}}\) - This option incorrectly features a \(r^{3}\) in the denominator.(d) \(E = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q^{2}}{r^{2}}\) - This option has \(Q^{2}\) instead of \(Q\).The correct answer, therefore, is option (a).

Step 4: Conclusion

Based on our understanding and application of Gauss's Law, the correct formula for the electric field of a point outside the sphere is found in option (a). Thus, we conclude that the electric field at \(r > R\) is\[ E = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{2}} \].

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux passing through a closed surface to the charge enclosed by that surface. This law is vital for solving problems involving electric fields, especially when dealing with symmetrically charged objects like spheres. To apply Gauss's Law, consider an imaginary closed surface, known as a Gaussian surface, encompassing the charge distribution of interest.

According to Gauss's Law, the electric flux \( \Phi \) is given by the equation:

\[ \Phi = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_{0}} \]

Here, \( \mathbf{E} \) is the electric field, \( d\mathbf{A} \) is a differential area on the surface, and \( Q_{\text{enc}} \) is the enclosed charge. For a point outside a charged sphere, where the enclosed charge is the entire charge \( Q \) of the sphere, Gauss's Law simplifies the calculation. The symmetry of a sphere allows us to consider the electric field as constant over its surface, leading to the effective calculation of the electric field using this principle.

Point Charge

A point charge refers to an idealized model of a charge with an infinitesimally small size, making calculation and theoretical analysis simpler. In nature, we do not find true point charges, as every charged particle has a finite size. However, modeling a charge as a point helps in simplifying the study of electric fields and forces.

For a point charge \( Q \) in space, the electric field it generates diminishes with the square of the distance \( r \) from the charge. This is often described by the formula:

\[ E = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{r^{2}} \]

This expression shows that the electric field radiates outward from the charge in all directions and decreases as one moves further away. In the case of a charged sphere, when observing from a point outside the sphere, it behaves like a point charge placed at its center, as the distribution of the charge seems uniform from a distance.

Coulomb's Law

Coulomb's Law is a crucial principle in electrostatics, describing the force between two point charges. This law highlights how both the magnitude and direction of the electrostatic force between charged objects are governed by their charges and the distance separating them.

According to Coulomb's Law, the force \( F \) between two charges \( q_1 \) and \( q_2 \), situated a distance \( r \) apart, is given by:

\[ F = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{|q_1 \cdot q_2|}{r^{2}} \]

This formula implies that the force is inversely proportional to the square of the distance between the charges and directly proportional to the product of the charges. Like gravitational forces, electrostatic forces deduced from Coulomb's Law are central and conservative. Moreover, when applied to a charged sphere, this principle helps to deduce the electric field, assuming the sphere's charge acts as though it is concentrated at one point at its center. Thus, even for complex shapes, Coulomb's Law aids in determining how charges interact across distances.