Figure 28.20 a shows a Helmholtz coil used to generate...

Figure 28.20 a shows a Helmholtz coil used to generate uniform magnetic fields. Suppose the Helmholtz coil consists of two sets of coaxial wire loops with 15 turns of radius $R=75.0 \mathrm{~cm},$ which are separated by $R$, and each coil carries a current of 0.123 A flowing in the same direction. Calculate the magnitude and the direction of the magnetic field in the center between the coils.

Key Concepts

Biot–Savart Law

The Biot–Savart law is a fundamental principle that relates current elements to the magnetic fields they produce. It is essential for calculating the magnetic field at any point in space due to a distribution of currents, such as those in the loops of a Helmholtz coil.

Helmholtz Coil

A Helmholtz coil consists of two identical circular loops placed symmetrically along a common axis with a separation equal to their radius. This design creates a region near the center where the magnetic field is very uniform, making it ideal for experiments that require a consistent magnetic field over a volume.

Magnetic Field of a Circular Loop

The magnetic field produced by a current-carrying circular loop can be determined using a formula derived from the Biot–Savart law. This formula calculates the field along the axis of the loop and serves as the fundamental building block for finding the field contribution from each coil in a Helmholtz configuration.

Superposition Principle

The superposition principle states that the total magnetic field resulting from multiple current sources is the vector sum of the individual fields produced by each source. This concept is critical in helmholtz coil configurations where the fields from two separate coils are combined to yield a uniform magnetic field between them.

Transcript

00:02 Hi, in the given problem, this is the line joining the centers of the two coils.

00:12 Then these are the two coils, identical coils, both of them are.

00:31 So the direction of current in both of the coils is same and that is clockwise.

00:40 The identical current, i1 is here, i2 is here, but both the currents are identical.

00:49 I1 is equal to i2 is equal to 0 .1, 2, 3 ampere.

00:57 Now we have to find the magnetic field at a point midway between these two coils.

01:06 Suppose the point to be p.

01:08 Now the radius of each coil is given as r and the gap between them is also given as r.

01:19 So r1 is equal to r2, let it be r which is equal to 75 cm or we can say this is 0 .75 meter.

01:32 So the distance of observation point from both the coils that will also be same which will be half of r means that will be 0 .375 meter.

01:53 Now as the direction of current in this loop the first loop is clockwise as seen from point p so it will be behaving like south pole for this point so the magnetic field here will be entering into it.

02:10 So let it be b1 due to the loop 1.

02:16 Now as seen from this point p, the face of the coil, in this face of the coil, the direction of current will be counterclockwise.

02:24 As for us, we are looking at the coil here from right hand side, but the point p will be looking the coil from opposite side...

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