Question

The magnetic field on the axis of a circular current loop (Eq. 5.41) is far from uniform (it falls off sharply with increasing z). You can produce a more nearly uniform field by using two such loops a distance d apart (Fig. 5.59). (a) Find the field (B) as a function of z, and show that ∂B/∂z is zero at the point midway between them (z = 0). (b) If you pick d just right, the second derivative of B will also vanish at the mid- point. This arrangement is known as a Helmholtz coil; it’s a convenient way of producing relatively uniform fields in the laboratory. Determine d such that ∂2 B/∂z2 = 0 at the midpoint, and find the resulting magnetic field at the center. √ [Answer:8μ0I/5 5R]

Answer 1

Answer:

Part b

$B = 8\frac{\mu_o I}{5\sqrt5 R}$

Explanation:

Part a)

Let the radius of the coil is R and magnetic field on its axis is given as

$B = \frac{\mu_o I R^2}{2(z^2 + R^2)^{3/2}}$

now we know that two coils are identical and we need to find magnetic field between two coils

so we will have net magnetic field given as

$B = \frac{\mu_o I R^2}{2(z^2 + R^2)^{3/2}} + \frac{\mu_o I R^2}{2((d-z)^2 + R^2)^{3/2}}$

Now we know that magnetic field is maximum at position

$\frac{dB}{dz} = 0$

so we will have

$z - (d - z) = 0$

$z = \frac{d}{2}$

so it will be at mid point of two coils

Part b)

Now we know that

$\frac{d^2B}{dz^2} = 0$

so we will have

$d = R$

now magnetic field is given as

$B = 2\frac{\mu_o I R^2}{2(z^2 + R^2)^{3/2}}$

put z = 0.5 R

$B = 8\frac{\mu_o I}{5\sqrt5 R}$