Two concentric circular wire loops in the same plane each...

Two concentric circular wire loops in the same plane each carry a current. The larger loop has a current of 8.46 A circulating clockwise and has a radius of $6.20 \mathrm{cm}$. The smaller loop has a radius of $4.42 \mathrm{cm} .$ What is the current in the smaller loop if the total magnetic field at the center of the system is zero? [See Eq. ( $19-32$ ). $.$

Key Concepts

Magnetic Field of a Circular Current Loop

This concept describes how a current traveling in a circular wire loop generates a magnetic field at its center. The field is given by the formula B = (?0 * I) / (2R), where I is the current and R is the radius of the loop. This fundamental relationship, derived from the Biot–Savart law, is widely used to analyze and design systems involving loop magnets.

Superposition Principle in Magnetism

Magnetic fields are vector quantities, meaning they have both magnitude and direction. The superposition principle tells us that when there are multiple sources of magnetic fields, the overall field is the vector sum of the individual fields. This principle allows us to set conditions, such as having a zero net magnetic field at a given point by appropriately balancing the contributions from different loops.

Right-Hand Rule for Determining Field Direction

The right-hand rule is a convention used to determine the direction of the magnetic field generated by a current-carrying loop. By curling the fingers in the direction of the current, the thumb points in the direction of the magnetic field inside the loop. Understanding this rule is essential for predicting how different loops will interact, especially when their current directions are opposite.

Magnetic Field Cancellation Conditions

This concept involves analyzing and setting the conditions under which the magnetic fields from different sources cancel out completely at a specific point. In problems where the net magnetic field is required to be zero, the individual fields must not only have the same magnitude but also opposite directions. This balance is achieved by relating the currents and the geometrical parameters of the loops.

Transcript

00:01 In this question, we have two circular wire loops.

00:03 They are both in the same plane and carry currents.

00:06 The large loop has a current of 8 .46 amps.

00:10 We can see as it's drawn, it is labelled in a clockwise direction.

00:16 We also know that the larger radius is equal to 6 .2 times 10 to 2 metres, and r1, the smaller radius, is 4 .42 times 10 to the minus 2 meters.

00:26 We are told that the total magnetic field at the centre, so point p labeled here, is equal to zero.

00:33 What we want to find is the current in the smaller loop to make this possible.

00:38 So, what we can say is we know that the magnetic field at the centre of a circular loop of radius r is given by mu not, well, this is a constant, i over to r.

00:54 This would be multiplied by n if there were more turns in the loop.

00:57 However, we just have one, so this is the expression we're going to use.

01:01 What we can say now is we know that the right, from the right -hand grip rule, the magnetic field at the centre of a large loop is into the page.

01:11 Which means, in order for the net magnetic field to be equal to zero, it has to be in the opposite direction for the smaller loop, because the fields are anti -parallel.

01:22 So we can say that the current in the smaller loop is counterclockwise.

01:28 What we now know is that the magnetic field due to each must be equal in order for them to be zero to cancel out as they are in opposite directions.

01:37 So we can say that the magnetic field of the small loop is equal to the magnetic field of the large loop...

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