Question
A charge is distributed uniformly over a ring of radius ‘a’. Obtain an expression for the electric intensity E at a point on the axis of the ring. Hence show that for points at large distances from the ring, it behaves like a point charge.
✓
Answer
Net Electric Field at point $\text{P} =\int^{2\pi\text{a}}_{\circ}\text{dE}\cos\theta$
dE = Electric field due to a small element having charge dq
$ = \frac{1}{4\pi\varepsilon_\circ}\frac{\text{dq}}{\text{r}^{2}}$
Let $\lambda$ = Linear charge density
$ = \frac{\text{dq}}{\text{dl}}$
$\text{dq} = \lambda\text{dl}$
Hence $\text{E} = \int^{2\pi\text{a}}_{\circ}\frac{1}{4\pi\varepsilon_\circ}.\frac{\lambda\text{dl}}{\text{r}^{2}}\times\frac{x}{r} , $
where $\cos\theta =\frac{\text{x}}{\text{r}}$
$ = \frac{\lambda\text{x}}{4\pi\varepsilon_\circ\text{r}^{3}}(2\pi\text{a})$
$ =\frac{1}{4\pi\varepsilon_\circ}\frac{\text{Qx}}{(\text{x}^{2} + \text{a}^{2})^{\frac{3}{2}}}$
where total charge $\text{Q} = \lambda\times2\pi\text{a}$
At large distance i.e. x >> a
$\text{E}\simeq\frac{1}{4\pi\varepsilon_\circ}.\frac{\text{Q}}{\text{x}^{2}}$
This is the Electric field due to a point charge at distance x.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the angle of deviation suffered by the light ray shown in figure. The refractive index $\mu=1.5$ for the prism material.
→In defining the ideal gas temperature scale, it is assumed that the pressure of the gas at constant volume is proportional to the temperature T. How can we verify whether this is true or not? Are we using the kinetic theory of gases? Are we using the experimental result that the pressure is proportional to temperature?
Find electric field of an electric dipole at any point on the axis of dipole. Also write one special case.
Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).
(a) The radii of curvature of the faces of a double convex lens are 10 cm and 15 cm . Its focal length is 12 cm . What is the refractive index of glass?
(b) A convex lens has 15 cm focal length in air. What is focal length in water? (Refractive index of air-water 1.33 and Refractive index of air-glass $=1.5$ )
→(b) A convex lens has 15 cm focal length in air. What is focal length in water? (Refractive index of air-water 1.33 and Refractive index of air-glass $=1.5$ )
The specific heat capacities of hydrogen at constant volume and at constant pressure are 2.4cal-g-1 °C−1 and 3.4cal-g-1 °C-1 respectively. The molecular weight of hydrogen is 2g-mol-1 and the gas constant, R = 8.3 × 107 erg°C-1 mol-1. Calculate the value of J.
One end of a steel rod $(\text{K}=46\text{Js}^{-1}\text{m}^{-1}{^{\circ}}\text{C}^{-1})$ of length 1.0m is kept in ice at 0°C and the other end is kept in boiling water at 100°C. The area of cross section of the rod is 0.04cm2. Assuming no heat loss to the atmosphere, find the mass of the ice melting per second. Latent heat of fusion of ice $=3.36\times10^5\text{Jkg}^{-1}.$
→In the original Fizeau method, the light travelled 8.6km and then returned. What could be the difficulty if this distance is taken as 8.6m?
Two wires A and B of the same material and having same length, have their cross sectional areas in the ratio 1 : 6. What would be the ratio of heat produced in these wires when same voltage is applied across each?
What is nuclear binding energy?