Derivation of Work done to move a point charge from infinity to a point P.

 Work done to move a point charge from infinity to point p . 

Derivations class 12 physics

To understand the derivation of the work done to move a point charge from infinity to a point, we need to consider the electric field and potential due to a point charge. Let's assume we have a positive point charge, denoted as Q, and we want to move it from infinity to a point P.

Here's a step-by-step explanation of the derivation:

1. Start by considering the electric field due to the point charge. According to Coulomb's law, the electric field at any point due to a point charge is given by:

   E = (k * Q) / r^2

   where E is the electric field, k is Coulomb's constant (9 x 10^9 Nm^2/C^2), Q is the charge, and r is the distance from the charge.

2. The work done in moving a charge through an electric field is given by the equation:

   W = q * ΔV

   where W is the work done, q is the charge, and ΔV is the change in potential.

3. Now, let's consider an incremental displacement of the charge from a distance r to r + dr. The work done in moving this small distance can be written as:

   dW = F * dr

   where F is the force experienced by the charge.

4. The force experienced by the charge is given by:

   F = q * E

   substituting the value of electric field E from step 1.

5. Substituting the value of force in the expression for work done, we get:

   dW = (q * E) * dr

6. We can rewrite the electric field E as:

   E = (k * Q) / r^2

   Substituting this in the equation for work done, we have:

   dW = (q * (k * Q) / r^2) * dr

7. Now, we can integrate the expression for incremental work done over the entire distance from infinity (r = ∞) to the final position (r = r):

   ∫dW = ∫(q * (k * Q) / r^2) * dr

   where ∫ denotes the integral.

8. Integrating the equation, we get:

   W = q * (k * Q) * (∞ - r) / r^2

   or,

   W = (q * k * Q) / r

This is the expression for the work done to move a point charge from infinity to a point at a distance r. It depends on the product of the two charges (q and Q), Coulomb's constant (k), and the final distance (r).

Please note that the above derivation assumes that the charge Q is fixed and the charge q is the one being moved. Also, the derivation considers the movement of a positive charge, but the concept remains the same for negative charges with appropriate signs.