Infinite number of straight wires each carrying current I are equally placed as shown in the figure Adjacent wires have current in opposite direction Net magnetic field at point `P` is
.

Options

A. Zero
B. `(mu_0I)/(8pi)(log_e2)/(rsqrt3)`
C. `(mu_0I)/(4pi)(log_e2)/(rsqrt3)`
D. `(mu_0I)/(2pi)(log_e2)/(rsqrt3)`

Answer

D

Explanation

the magnetic field at O due to straight wire carrying current is
`vecB=(mu_0)/(4pi)I/a[sin 30^@+sin 30^@]hatk`
`=(mu_0)/(4pi)I/a[1/2+1/2]hatk=(mu_0)/(4pi)I/ahatk`

Total magnetic field at O due to current through all the straight parallel wires carrying currents
`vecB=B_1hatk+B_2(-hatk)+B_3(hatk)+B_4(-hatk)+...`
`B=B_1-B_2+B_3-B_4+...`
`=(mu_0)/(4pi)I/a[1-1/2+1/3-1/4+...]`
`=(mu_0)/(4pi)I/alog_e(1+1)=(mu_0)/(4pi)I/alog_e2`
Here, `cos 30^@=a/r` or `a=r cos 30^@=rsqrt3//2`
`:. B=(mu_0)/(4pi)(I)/((rsqrt3//2))log_e2=(mu_0)/(2pi)(I)/(rsqrt3)log_e2`