Name: Magnetic Flux
Uploaded: 2019-07-19
Duration: 6 min 44 s
Description: Magnetic flux is the total number of magnetic lines of force that are passing through a given surface. To learn more about this intriguing concept, watch the...

Question

Helmholtz coils consist of two large flat circular coils mounted with a common axis and at distance apart equal to their common radius. Show that this arrangement produces a uniform magnetic flux density over a small space midway between the coils. Derive an equation for this flux density when the coils have the same current in the direction to produce additive effect.

Accepted Answer

Consider a circular current-carrying coil carrying current I. The magnetic field at any axis at a distance $x$ from the centre of the coil is

$B = \frac{μ_{0} {IR}^{2} N}{2 {(x^{2} + R^{2})}^{\frac{3}{2}}}$ .............................................(1)

where $N$ = number of coils, $R$ = radius of the coil

From equation- $(1)$ it is seen that the magnetic field varies with varying a distance from centre to at any point along the axis.

Helmholtz found that when two identical co-axial circular current-carrying coil carrying current in the same direction are placed at a distance which is equal to the radius of the coil, the magnetic field due to coil is uniform.

The magnitude of the field at the midpoint (at distance $x = \frac{R}{2}$ ) will be,

$B = \frac{μ_{0} {IR}^{2} N}{2 {(R^{2} + {(\frac{R}{2})}^{2})}^{\frac{3}{2}}} = \frac{μ_{0} {IR}^{2} N}{2 R^{2} {(1 + (\frac{1}{4}))}^{\frac{3}{2}}} = \frac{μ_{0} IN}{2 R {(\frac{5}{4})}^{\frac{3}{2}}}$

As the current is in the same direction, the total field will be additive i.e twice of this.

The total field at the midpoint will be

$2 B = \frac{μ_{0} IN}{R {(\frac{5}{4})}^{\frac{3}{2}}} = 0.72 \frac{μ_{0} IN}{R}$ .

Important Questions on Magnetic Effect of Electric Current

Learn and Prepare for any exam you want !