Electric field due to circular loop of charge | Electricity and Magnetism

Sacademy May 15, 2023 1 min read

Electric field due to circular loop of charge

Electric field

The space around a charged particle in which another charge experience a force is known as electric field.
The source of electric field is either a charge or a time varying magnetic field.
If the value of electric field does not change with time, then it will be uniform electric field, otherwise it will be non-uniform electric field.

dq = λ dl ⇒ dq = (q / 2πa) dl

$dE = |d\vec{E}|=\frac{kdq}{(CP)^{2}}$

$or\;\;\;\;\;dE=\frac{kqdl}{2\pi ax^{3}}\;\;\;\;\;\left ( \because dq=\frac{q}{2\pi a}dl \right )$

$E=\int dE_{x}+\int dE_{y}$

$\int dE_{y}=0$

and dE_x = dE cos θ

$\therefore \;\;\;\;\;E=\int dEcos\theta \;\;\;\;\; along PX$

In ΔCPO, cos θ = r / x

$\therefore \;\;\;\;\;E=\int \left ( \frac{kqdl}{2\pi ax^{3}}\times \frac{r}{x} \right )=\frac{kqr}{2\pi ax^{3}}\int dl=\frac{kqr}{x^{3}}$

$or \;\;\;\;\;E=\frac{kqr}{(r^{2}+a^{2})^{3}}\;\;\;\;\;along PX$

{since x² = r² + a²}

When P lies at the centre of the loop i. e., r = 0, then E = 0
When P lies very far from the centre of the loop i. e., r >> a, then E = kq / r²
In this case circular loop behaves as a point charge.

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