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Homework Statement
a)recall that electric potential is a scalar quantity. For a circular ring of radius, R', carrying charge, Q, what is the electric potential at a height,y, above the center of the ring?
b)Use your above answer to determine the electric potential at a height,y, above the center of a uniformly charged disk of radius, R.
c)if y>>R, use the binomial expansion (keeping terms up to and including R^{2}/y^{2} ) to show that the electric potential now resembles that of a point charge
Homework Equations
V=q/(4πε_{0}r)
The Attempt at a Solution
a)V=q/(4πε_{0}r)
dv=dq/(4πε_{0})
See ring.jpg attached
V=∫dv=∫dq/(4πε_{0})=[1/(4πε_{0}r)]∫dq=q/4πε_{0}r=q/4πε_{0}(R'^{2}+y^{2})^{1/2}
b)See disk.jpg attached
V=kq/(R'^{2}+y^{2})^{1/2}
dV=kdq/(R'^{2}+y^{2})^{1/2}
(dq/Q)=(2πR'dr/πR^{2})→dq=2QR'dr/R^{2}
The limits on the below integral are from 0→R until stated differently
V=∫dV=∫(k2QR'dr)/[R^{2}(R'^{2}+y^{2})^{1/2}]=(kQ/R^{2})∫2R'dr/(R'^{2}+y^{2})^{1/2}
Usub:
U=R'^{2}+y^{2}
dU=2R'dr
dr=dU/2R'
New Limits:
when r=0, U=y^{2}
when r=R, U= R^{2}+y^{2}
after taking integral:
(KQ/R^{2}) [u^{1/2}]evaulated from y^{2}→R^{2}+y^{2}=(KQ/R^{2})[(R^{2}+y^{2})y]
c) This is the part Im confused on... not sure where to start.
I know if y>>R then my above answer turns into
(KQ/R^{2})[y^{2}y] but other than that not sure where to go...
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