Question

A long uniformly charged wire with linear charge density $\lambda=\varepsilon_{0} C / m$ is placed in the form of a spiral of outer radius $R=1\, m$ and having a total $N=160$ turns as shown in the figure. A dipole of dipole moment $|\vec{p}|=0.1 \,C m$ is placed along the $x$-axis at a point $(a, 0,0)$, where $a=\sqrt{3} m$. Find the magnitude of force (in $N$ ) experienced by the dipole.

Answer 1

$R$ length has turn $=N$
$d r$ length has turn $=\frac{N}{d r}$
$d q=\lambda \times 2 \pi r \frac{N}{R} d r$
$\frac{d q}{2 \pi r d r}=\frac{\lambda N}{R}=$ Constant $=\sigma$
Force $=\left(p \frac{d E}{d r}\right)=\frac{\lambda N p R}{2 \varepsilon_{0}\left(R^{2}+a^{2}\right)^{3 / 2}}$
$F=\frac{\varepsilon_{0} \times 160 \times 0.1 \times 1}{2 \varepsilon_{0} \times(1+3)^{3 / 2}}=1$