Variation of Electric Field and Potential Due to a Charged Sphere

Setup: A sphere of radius R carries total charge q, with surface charge density σ.

• As r → 0, E stays 0 (not infinite — no charge enclosed).
• As r → ∞, E → 0 (inverse-square decay).

• As r → 0, V does NOT go to infinity — it stays capped at the surface value.
• As r → ∞, V → 0.

Learning Objective: Express work done moving a charge through a field as a line integral.

A source charge +q sits at origin O. A test charge q₀ at point P experiences force:

Since E = F/q₀:

• \[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat{r}\]   ...(1)

For a small displacement dl from P to Q, work done: dW = E · dl

Total work done moving from A to B:

This integral is called the line integral of the electric field between A and B.

Analogy: Like adding up small bits of effort while walking an uneven hiking trail — you sum the effort at every step regardless of the path's shape.

Learning Objective: Prove that work done moving a charge depends only on start/end points, not the path — i.e., the field is conservative.

To move test charge q₀ without acceleration, apply opposing force: F = −q₀E

Work done moving along any path from A to B (using geometry: dr = dl cos θ):

Result depends only on r_A and r_B — not on the path shape (straight or curved).

Dividing by q₀:

Analogy: Just like gravitational PE — climbing a hill via a steep shortcut or a long winding trail takes different effort along the way, but the net height change is identical.

Learning Objective: Show the total work done moving a charge around any closed loop is zero.

Closed path: A → P → B → Q → A

Along path P (A → B):

Along path Q (B → A):

Adding both → sum = 0, so for the full closed loop:

Analogy: Walking a full loop back to your starting point on a hill — your net height change is zero, regardless of the loop's shape.