The inductance L of a singlecoil system is given in the foll

The inductance L of a single-coil system is given in the following quadratic function of x: L = a_0 + a_1 x - a_2 x^2 in Henry, where a_0 = 2, a_1 = 10, a_2 = 250 if x is in meters. Determine x such that L reaches its maximum value. What is the magnetic force f_m at the above x in (a)? If now the magnetic force f_m is balanced by a spring with the spring constant k = 100 N/m, find x if the coil current i = 2A.

Solution

(a) L will be maximum at which dL/dx=0.

dL/dx=0

a1-2a2*x=0

X= a1/(2a2)= 0.02 m

( b) fm= magnetic force = i^2/2*dL/dx

Where i= current

fm= (a1-2a2*x)*i^2/2

At x= 0.02 m, fm=0.

(C) fs= spring froce = kx= 100x

fm= 2(a1-2a2x)= 2(10-500x)

fm=fs

2(10-500x)= 100x

X= 10/550= 0.018 m