What is work done in taking a positive charge from one point to other (i) inside a positively charged metallic sphere and (ii) Outside the sphere.

To understand the work done in moving a positive charge within and outside a positively charged metallic sphere, we need to consider the electric field created by the sphere and how it affects the charge's movement. Let's break this down into two parts: moving the charge inside the sphere and moving it outside.

Work Done Inside the Sphere

When you move a positive charge from one point to another inside a positively charged metallic sphere, the situation is quite interesting. Inside a conductor in electrostatic equilibrium, the electric field is zero. This means that there are no electric forces acting on the charge as you move it around within the sphere.

• No Electric Field: Since the electric field is zero, there is no force acting on the charge.
• Work Calculation: Work done (W) is calculated using the formula W = F × d, where F is the force and d is the distance moved. Here, F = 0, so W = 0.

Therefore, regardless of how far you move the positive charge within the sphere, the work done is zero. This is a fundamental property of conductors in electrostatic equilibrium.

Work Done Outside the Sphere

Now, let’s consider moving the same positive charge outside the positively charged metallic sphere. In this case, the situation changes significantly. The electric field outside a charged sphere behaves as if all the charge were concentrated at its center. This means that the electric field (E) is not zero, and it varies with distance from the sphere.

• Electric Field Outside: The electric field (E) at a distance r from the center of the sphere can be calculated using Coulomb's law: E = k * Q / r², where k is Coulomb's constant and Q is the total charge of the sphere.
• Force on the Charge: When you move the positive charge in this electric field, it experiences a force given by F = qE, where q is the charge being moved.
• Work Calculation: The work done in moving the charge from one point to another outside the sphere can be calculated by integrating the force over the distance moved. If you move the charge from a distance r1 to r2, the work done is W = ∫(F)·dr from r1 to r2.

In this scenario, the work done will be positive if you move the charge against the electric field (toward the sphere) and negative if you move it in the direction of the field (away from the sphere). The specific value of work done will depend on the distances involved and the amount of charge being moved.

Summary

In summary, when moving a positive charge:

• Inside a positively charged metallic sphere, the work done is zero due to the absence of an electric field.
• Outside the sphere, the work done depends on the electric field and the distance moved, resulting in a non-zero value.

This understanding of electric fields and work done is crucial in fields like electrostatics and electrical engineering, where the behavior of charges in different environments is fundamental to designing circuits and understanding electrical interactions.