The region between two concentric spheres of radii a and b, respectively (see the figure),
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By symmetry electric field in the region between two shells is radial . Consider a shell of radius r . Flux through this shell is :
ϕ=∮E→.dS→ϕ=∮E→.dS→
ϕ=E4πr2=qinc∈0ϕ=E4πr2=qinc∈0
Net charge between r=a to r=r would be
Q=∫raρ(4πr2)dr=∫raCr4πr2drQ=∫arρ(4πr2)dr=∫arCr4πr2dr
=4πC2[r2]ra=4πC2[r2]ar
Q=2πC(r2−a2)Q=2πC(r2−a2)
E4πr2=Q+q∈0E4πr2=Q+q∈0
E=k(2πC(r2−a2)+q)r2E=k(2πC(r2−a2)+q)r2
E=2πkC+kr2(−2πCa2+q)E=2πkC+kr2(−2πCa2+q)
Since E is constant . It should be independent of r
q−2πCa2=0q−2πCa2=0
C=q2πa2.
ϕ=∮E→.dS→ϕ=∮E→.dS→
ϕ=E4πr2=qinc∈0ϕ=E4πr2=qinc∈0
Net charge between r=a to r=r would be
Q=∫raρ(4πr2)dr=∫raCr4πr2drQ=∫arρ(4πr2)dr=∫arCr4πr2dr
=4πC2[r2]ra=4πC2[r2]ar
Q=2πC(r2−a2)Q=2πC(r2−a2)
E4πr2=Q+q∈0E4πr2=Q+q∈0
E=k(2πC(r2−a2)+q)r2E=k(2πC(r2−a2)+q)r2
E=2πkC+kr2(−2πCa2+q)E=2πkC+kr2(−2πCa2+q)
Since E is constant . It should be independent of r
q−2πCa2=0q−2πCa2=0
C=q2πa2.
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