Calculate potential on the axis of a ring due to charge Q uniformly distributed along the ring of radius R.
Let us consider a ring of radius R having charge +Q distributed uniformly. Also a point P at distance z on its axis passing through centre 0 and perpendicular to plane of ring. Again consider an element of ring at S of length di having charge dq and SP is equal to r. Then potential energy due to element to r. Then potential energy due to element dl at P, dV=r−kdq where k=4πε01 Charge on 2πR length of ring =Q Charge on dl length of ring =2πRQdl So potential due to element di at P dV==2πR−k⋅Q⋅dl Integrating over a ring the potential at P,VP 0∫vdVp=0∫2πR2πRrkQdl where r=R2+z2 Vp=2πRR2+z2kQ2πR=4πε0R2+z2Q Charge on 2Charge on 2\piR length of ring = Charge on dl length of ring =2πRQdl So potential due to element di at P dV==2πR−k⋅Q⋅dl Integrating over a ring the potential at P,VP ∫0vdVp=∫02mR2πRrkQa1 where r=R2+z2 Vp=2πRR2+z2kQ2πR=4πε0R2+z2Q
Let us consider a ring of radius R having charge +Q distributed uniformly. Also a point P at distance z on its axis passing through centre 0 and perpendicular to plane of ring. Again consider an element of ring at S of length di having charge dq and SP is equal to r. Then potential energy due to element to r. Then potential energy due to element dl at P, dV=r−kdq where k=4πε01 Charge on 2πR length of ring =Q
Let us consider a ring of radius R having charge +Q distributed uniformly. Also a point P at distance z on its axis passing through centre 0 and perpendicular to plane of ring. Again consider an element of ring at S of length di having charge dq and SP is equal to r. Then potential energy due to element to r. Then potential energy due to element dl at P, dV=r−kdq where k=1/4πε0
