Problem 56 - Master Resource Book in JEE Main Physics

Question

An electric dipole is placed at the centre of a hollow conducting sphere. Which of the following option is correct? (a) Electric field is non-zero anywhere on the sphere (b) The flux of electric field is zero through the sphere (c) Options (a) and (b) both are true (d) No option is correct

Step-by-Step Solution

Answer

Option (b) is correct: The flux of electric field is zero through the sphere.

Step 1: Understanding Gauss's Law

Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, this is expressed as \( \Phi = \frac{Q_{ ext{enclosed}}}{\varepsilon_0} \). If there is no charge enclosed, the electric flux \( \Phi \) is zero.

Step 2: Analyzing the Charge Distribution

In this scenario, an electric dipole is placed at the center of a hollow sphere. A dipole consists of two equal and opposite charges, which means the net charge is zero. Therefore, the charge enclosed by the sphere is zero.

Step 3: Applying Gauss's Law to the Sphere

Since a dipole (net charge zero) is enclosed by the sphere, according to Gauss's Law, the electric flux through the hollow conducting sphere is zero, as given by \( \Phi = 0 \). Thus, statement (b) is correct.

Step 4: Evaluating Electric Field on the Sphere

Even though the flux through a closed surface is zero when a dipole is inside, the electric field is not necessarily zero everywhere on the surface of the sphere. The electric field due to a dipole is non-uniform and can be present on the surface, but because it integrates to zero over the entire surface, this doesn't contradict the previous step.

Step 5: Conclusion

Based on Steps 1 to 4, statement (b) is correct because the net electric flux through the sphere is zero due to the net zero charge enclosed. Option (a) can be technically true as the field may not be zero everywhere, but option (b) is definitively correct.

Key Concepts

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a certain distance. Imagine a positive charge and a negative charge connected by an invisible line. This simple setup gives rise to several interesting properties. The most notable is the dipole moment, denoted by \( \overrightarrow{p} \), which is a vector quantity. Its magnitude is the product of the charge value \( q \) and the distance \( d \) between the charges: \( p = q \times d \).
Dipoles are significant in electric fields, affecting the field lines around them. At points along the axis extending through both charges, the field lines are straightforward, but the situation is more complex in other regions. The presence of a dipole influences fields, especially when placed within or around other charges or conductors, such as a hollow conducting sphere.

Electric Flux

Electric flux refers to the total amount of electric field passing through a surface. You could think of electric flux as the net number of field lines passing through a given area. It is mathematically expressed as \( \Phi = \int \overrightarrow{E} \cdot d\overrightarrow{A} \). This means it depends on the strength of the electric field \( \overrightarrow{E} \) and the orientation and area of the surface \( d\overrightarrow{A} \).
Gauss's Law connects electric flux to the concept of enclosed charge. According to this law, the flux through any closed surface is directly proportional to the enclosed charge. In scenarios where no net charge is enclosed, such as around an electric dipole centered in a sphere, the electric flux is zero. This doesn't mean there are no field lines, but rather their net effect through the closed surface cancels out.

Conducting Sphere

A conducting sphere can perfectly shield its interior from external electric fields, thanks to the mobile charges within the conductor. When placed in an electric field, the charges on the sphere move until equilibrium is reached, meaning the electric field inside becomes zero.
However, when an electric dipole is placed at the center of a hollow conducting sphere, the scenario is different. The net charge enclosed remains zero due to the dipole's nature, resulting in zero electric flux through the sphere. But it's crucial to note that the electric field is not necessarily zero at every point on the surface. Due to the dipole's influence, the field can vary in intensity and direction.

Enclosed Charge

The concept of enclosed charge is central to understanding electric flux through closed surfaces. Enclosed charge refers to the total charge contained within a specific surface or volume. In the context of Gauss's Law, we focus on the net charge, which is pivotal in determining the electric flux.
Even if charges within a surface create complex electric fields, what matters for electric flux is just the net charge enclosed. For instance, an electric dipole presents zero net enclosed charge because it consists of a pair of opposite but equal charges. This zero net charge means, per Gauss's Law, that the electric flux through a closed surface, like a hollow conducting sphere around the dipole, will be zero, exemplifying how electric flux is intricately tied to charge containment, not the complexity of the arrangement.