Question

Charge is uniformly distributed around a ring of radius R and the resulting electric field magnitude E is measured along the ring's central axis (perpendicular to the plane of the ring). At what distance from the ring's center is E maximum? (Use any variable or symbol stated above as necessary.)

Answer 1

`x=(r)/(\sqrt2)`

Step-by-step explanation:

Given that

Radius =r

Electric filed =E

Q=Charge on the ring

The electric filed at distance x given as

`E=K(Q)/((r^2+x^2)^(3/2))`

For maximum condition

`(dE)/(dx)=0`

`E=K{Q}{(r^2+x^2)^(-3/2)}`

`(dE)/(dx)=K{Q}{(r^2+x^2)^(-3/2)}-(3)/(2)* 2* x* K{Q}{(r^2+x^2)^(-5/2)}`

For maximum condition

`(dE)/(dx)=0`

`K{Q}{(r^2+x^2)^(-3/2)}-(3)/(2)* 2* x* K{Q}{(r^2+x^2)^(-5/2)}=0`

`r^2+x^2-3x^2=0`

`x=(r)/(\sqrt2)`

At
`x=(r)/(\sqrt2)` the electric field will be maximum.