An inducatane L and a resistance R are connected in series with a battery of emf epsilon. Find the maximum rate at which the energy is stored in the magnetic field.

Options

A. `(epsilon^(2))/(4R)`
B. `(epsilon^(2))/(2R)`
C. `(2R)/(epsilon)`
D. `(4R)/(epsilon)`

Answer

A

Explanation

The energy stored in the magnetic field at tiem t, is
`U=(1)/(2)Li^(2)=(1)/(2)Li_(0)^(2)(1-e^(-t//tau))^(2)`
The rate at which the energy is stored is
`P=(dU)/(dt)`
`=Li_(0)^(2)(10e^(-t//tau))(-e^(-t//tau))(-(1)/(tau))`
`=(Li_(0)^(2))/(tau)(e^(-t//tau)-e^(-2t//tau)) " " ` ...(i)
This rate will be maximum, when
`(dP)/(dt)=0`
`rArr (-(1)/(tau)e^(-t//tau)+(2)/(tau)e^(-2t//tau))=0`
`rArr e^(-t//tau)=(1)/(2)`
Putting in Eq. (i), we have
`P_("max") =(Li_(0)^(2))/(tau)((1)/(2)-(1)/(4))`
`=(L epsilon^(2))/(4R^(2)((L)/(R)))`
`=(epsilon^(2))/(4R)`