Question 3: Electrostats & Capacitance

Question

A half ring of radius $$R$$ has a charge of $$\lambda$$ per unit length. The electric force on $$1 \mathrm{C}$$ charged placed at the centre is

Accepted Answer

As, $$R$$ be the radius of the ring. Consider a small strip of length $$d l$$ having charge $$d q$$ lying at an angle $$	heta$$.
$$d l=R d 	heta$$
Charge on $$d l=\lambda R d 	heta$$
Force at $$1 \mathrm{~C}$$ due to $$\mathrm{dl}$$
$$=\frac{k \lambda R d 	heta}{R^2}=\frac{k \lambda}{R} d 	heta=d F$$
We need to consider only the component $$d F \cos 	heta$$, as the component $$d F \sin 	heta$$ will cancel out because of the symmetrical element $$d l$$.
The total force on $$1 \mathrm{C}$$ is
$$\begin{aligned}
F & =\int_\limits{-\pi / 2}^{\pi / 2} d F \cos 	heta \
& =\frac{k \lambda}{R} \int_\limits{-\pi / 2}^{\pi / 2} \cos 	heta d 	heta \
& =\frac{k \lambda}{R} 	imes 2=\frac{2 k \lambda}{R}
\end{aligned}$$

Answer

zero

Answer

$$\frac{k \lambda}{R}$$

Answer

$$\frac{2 k \lambda}{R}$$

Answer

$$\frac{k \pi \lambda}{R}$$

Electrostats & Capacitance

A half ring of radius $R has a charge of \lambda per unit length. The electric force on 1 \mathrm{C}$ charged placed at the centre is

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