Question

Three isolated metal spheres A, B, C have radius R, 2R, 3R respectively, and same charge Q. UA, UB and UC be the energy density just outside the surface of the spheres. The relation between UA, UB and UC is

Answer 1

The energy density just outside the surface of a charged sphere is given by the formula:
UA = $\frac {1}{2}$ε₀E².
The electric field outside a uniformly charged sphere is given by:
E = $\frac {kQ}{r^2}$
Let's consider the three spheres A, B, and C with radii R, 2R, and 3R, respectively.
For sphere A (radius R):,ṁ
EA = $\frac {kQ}{R^2}$
UA = $\frac {1}{2}$ε₀$(\frac {kQ}{R^2})^2$ = $\frac {1}{2}$ε₀$\frac {k^2Q^2}{R^4}$
For sphere B (radius 2R):
EB = $\frac {kQ}{(2R)^2}$ = $\frac {kQ}{4R^2}$
UB = $\frac {1}{2}$ε₀$(\frac {kQ}{4R^2})^2$ = $\frac {1}{2}$ε₀$\frac {k^2Q^2}{16R^4}$ =$\frac {1}{8}$ε₀$\frac {k^2Q^2}{R^4}$
For sphere C (radius 3R):
EC = $\frac {kQ}{(3R)^2}$ = $\frac {kQ}{9R^2}$
UC = $\frac {1}{2}$ε₀$(\frac {kQ}{9R^2})^2$ = $\frac {1}{2}$ε₀$\frac {kQ}{81R^4}$ = $\frac {1}{18}$ε₀$\frac {k^2Q^2}{R^4}$
Therefore, the relation between UA, UB, and UC is:
UA : UB : UC = 1 : $\frac {1}{8}$ : $\frac {1}{18}$
UA : UB : UC = 18 : 2 : 1.
Hence, the relation between UA, UB, and UC is 18 : 2 : 1 i.e. UA>UB>UC