Two isolated metal spheres of radii r1 and r2 carry charges . . .

Question : Two isolated metal spheres of radii r1 and r2 carry charges q1 and q2 respectively. The spheres are brought into contact and then separated. Find the common potential of the spheres after contact.


Doubt by Riddhi

Solution : 
Total charge q=q1+q2

After the contact both the sphere will have same potential but their changes will be altered let their new charge will be q1' and q2'

V1=V2
kq1'/r1=kq2'/r2
q1'/r1=q2'/r2
q1'/q2'=r1/r2

As per the law of conservation of charges
q1+q2=q1'+q2'

q1'=r1(q1+q2)/(r1+r2)
q2'=r2(q1+q2)/(r1+r2)

V1=k(q1')/r1
V1=k[r1(q1+q2)/(r1+r2)]/r1
V1=k(q1+q2)/(r1+r2)

Similary 
V2=k(q1+q2)/(r1+r2)

VC=V1=V2=k(q1+q2)/(r1+r2)

Pro Tip : We can also solve this question by using the formula for capacitance of spherical conductor. (C=4πε₀R) and then using the formula for common potential.