Derivation of electric potential due to a point charge is?

Question

Answer

[K×(charge)/r]•here k is constant, and r is the distance between the point charge and the source

Answer

V = k q /r
where,
V=Potential due to that point
q= Charge {with sign}
r= distance
k = dielectric constant [ 1/ 4π epsilon naught]

Answer

Answer

Derivation of Electric Potential due to a Point Charge:

The electric potential, also known as voltage, at a point in the electric field created by a point charge can be derived using the concept of work done. Let's break down the derivation into the following steps:

Coulomb's Law:
Coulomb's law states that the force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:

F = k * (q1 * q2) / r^2

Where F is the force, q1 and q2 are the magnitudes of the charges, r is the distance between them, and k is the electrostatic constant.

Electric Field:
The electric field, E, at a point in space due to a point charge, q, is defined as the force experienced by a positive test charge, q0, placed at that point, divided by the magnitude of the test charge. It can be given by:

E = F / q0 = k * (q / r^2)

Definition of Electric Potential:
The electric potential, V, at a point in space is defined as the work done per unit positive charge in bringing a positive test charge, q0, from infinity to that point without acceleration. Mathematically, it can be expressed as:

V = W / q0

Where W is the work done.

Derivation:
To derive the electric potential due to a point charge, we consider the work done in moving a test charge, q0, from infinity to a point at a distance r from the charge.

- As the test charge is moved, the electric field does work against the force of the charge, q, creating a potential energy.
- The work done, W, can be calculated by integrating the force over the distance of displacement, which is given by:
W = ∫ F · dr = ∫ (q0 * E) · dr
- Since the electric field, E, is constant in magnitude and direction, it can be taken out of the integral, resulting in:
W = q0 * ∫ E · dr
- The integral of E · dr is the change in electric potential, V, from infinity to the point, and can be expressed as:
W = q0 * (V - 0)
W = q0 * V
- Substituting back into the previous equation, we have:
q0 * V = q0 * ∫ E · dr
V = ∫ E · dr
- Integrating E · dr gives the potential difference between the two points, which can be written as:
V = ∫ (k * q / r^2) · dr
V = k * q * ∫ (1 / r^2) · dr
V = k * q * (-1 / r)
V = - k * (q / r)

Conclusion:
The derived electric potential due to a point charge is given by V = - k * (q / r), where V is the electric potential, k is the electrostatic constant, q is the charge, and r is the distance from the charge. This equation represents

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