Figure 29-73 a shows a length of wire carrying a current i...

Figure $29-73 a$ shows a length of wire carrying a current $i$ and bent into a circular coil of one turn. In Fig. $29-73 b$ the same length of wire has been bent to give a coil of two turns, each of half the original radius. (a) If $B_{a}$ and $B_{b}$ are the magnitudes of the magnetic fields at the centers of the two coils, what is the ratio $B_{b} / B_{a} ?$ (b) What is the ratio $\mu_{b} / \mu_{a}$ of the dipole moment magnitudes of the coils?

Key Concepts

Magnetic Field of a Current Loop

The magnetic field at the center of a circular current loop is given by the expression B = ??i/(2r), where ?? is the permeability of free space, i is the current, and r is the radius of the loop. This formula shows that for a given current, the magnetic field strength is inversely proportional to the coil’s radius, a key principle when analyzing how changes in coil geometry affect the magnetic field.

Coil Geometry and Wire Length

When a fixed length of wire is used to form a coil, the dimensions of the coil, such as its radius and circumference, are directly related to that length. For instance, if the wire is bent into a single loop, the circumference determines the radius. If it is divided into multiple loops, each loop’s circumference becomes smaller, thereby reducing its radius. This concept is crucial for understanding how the magnetic field and area of each turn will vary with the geometry.

Multi-turn Coils

In a coil with multiple turns, the total magnetic field at the center is the superposition of the fields generated by each turn. Specifically, if the individual coils do not interfere destructively, the fields add algebraically, leading to a total magnetic field that is proportional to the number of turns. This linearity makes multi-turn coils useful for increasing the magnetic field without increasing the current.

Magnetic Dipole Moment

The magnetic dipole moment of a current-carrying loop is defined as ? = iNA, where i is the current, N is the number of turns, and A is the area of a single loop. This quantity characterizes the strength and orientation of the coil’s magnetic effect. Changes in the coil’s area or in the number of turns directly affect the dipole moment, and consequently the behavior of the coil in an external magnetic field.

Transcript

00:01 Hi in the given problem a given wire is first of all bent in the form of a single loop then the same wire is bent in the form of two loops so definitely the size of loop will decrease so here these are the two loops now this is the situation a this is is situation b.

00:34 Radius of first loop let it be r a and there let it be r b.

00:41 Now in the first part of the problem suppose length of the wire is l then when only one turn is there then circumference of this loop will be equal to length of the wire means 2 pi r a is equal to l so we can say r a is equal to l by 2 pi and the magnetic field at the center of this coil will be using mu not i because the current carried by the conductors is same so this will be mu not i by 2 r for r this is r a so here it will be mu not i by two times of l by 2 pi canceling the in this two finally we get this b a to be equal to mu not pi i by l now in case of two turns of the same length of wire so when there are two turns now, y's of circumference 2 by r b, because this time the radius is r b, we will be having the same length l.

02:36 Hence we can say now the radius r b is equal to l by 4 pi.

02:43 So this time the magnetic field will be given by mu not into n into i because the number of turns are 2 so this is 2 i divided by 2 r b canceling this 2 and putting the value of this radius r b we get mu not i by l by 4 pi so finally this magnetic field b b comes out to be 4 mu not by i divided by l.

03:23 Hence the ratio of magnetic field produced at the center of second coil to that of the first coil will be 4 is 2 1 which is the answer for the first part of this problem.

03:42 Now, in the second part of the problem, we have to find the ratio of the magnetic moment...

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