Let P(r)=(Q*r)/∏R4 be the charge density distribution for solid sphere of radius R & total charge Q. for a point p,inside a sphere at a distance r1 from centre of sphere,the magnitude of electric field is?
This problem came in one of the competitive exams few years ago and it had four options.
The field at r1 distance will be only due to the total charge within this radius. There will be no net contribution in field due to the charges that are outside the radius r1.
Charge within r1 radius = \(\int\limits_0^{{r_1}} {\frac{{Qr}}
{{\pi {R^4}}} \times 4\pi {r^2}dr = \frac{{Q{r_1}^4}}
{{{R^4}}}} \)
This charge is contained in a sphere of radius r1. To find the field at r1, this sphere can be treated as particle.
\(E = \frac{1}
{{4\pi { \varepsilon _0}}}\frac{{\frac{{Q{r_1}^4}}
{{{R^4}}}}}
{{{r_1}^2}} = \frac{1}
{{4\pi { \varepsilon _0}}}\frac{{Q{r_1}^2}}
{{{R^4}}}\)
This problem can be done using Gauss's law as well. Perhaps you may like to try it that way also.
sir,can you please tell me that why cant we use here the formula of electric field inside a sphere i.e E=rρ/3€?
The following points/questions can possibly lead you to figure out the answer yourself:
- \(\rho\) in your original question depends on r. Is the formula given by you valid for any variable \(\rho\) or constant \(\rho\)?
- Check the derivation of the formula given by you and compare it with the two methods mentioned by me earlier carefully. Just knowing the formula may not help always.
thank you sir,now i have understood my mistake.