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If you have some experience in electromagnetism and with vector calculus, prove that the magnetic forces, F12 and F21, between two steady current loops obey Newton's third law. [Hints: Let the two currents be I1 and I2 and let typical points on the two loops be r1 and r2. If dr1 and dr2 are short segments of the loops, then according to the Biot- Savart law, the force on dr1 due to dr2 is μ04πI1I2s2dr1×(dr2×s^) where s=r1r2. The force F12 is found by integrating this around both loops. You will need to use the " BACCAB " rule to simplify the triple product.]

Short Answer

Expert verified
The forces F12 and F21 are equal and opposite, satisfying Newton's third law.

Step by step solution

01

Understand the Problem

We need to prove Newton's third law for magnetic forces between two steady current loops, where the force between segments is given and related to the Biot-Savart law.
02

Biot-Savart Law Application

The force on a segment of loop 1 due to a segment on loop 2 is given by μ04πI1I2s2dr1×(dr2×s^), where s=r1r2. This is derived from the Biot-Savart law.
03

Simplify the Triple Product

Using the vector identity (a×(b×c))=(ac)b(ab)c, simplify dr1×(dr2×s^).
04

Express the Forces

The simplified expression for the force becomes dF12=μ04πI1I2s2((dr1s^)dr2(dr1dr2)s^).
05

Integrate Over the Loops

Integrate this expression over both loops to find the total forces F12 and F21.
06

Prove Newton's Third Law

By symmetry and integration limits, the forces on the segments of each loop will balance out, showing F12=F21, thus proving they satisfy Newton's third law.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Third Law
Newton's third law is a fundamental principle of physics. It states that for every action, there is an equal and opposite reaction. This means when one body exerts a force on a second body, the second body exerts a force of equal magnitude and opposite direction on the first body.
In the context of magnetic forces between two current loops, we express this with F12=F21. Here, F12 is the force exerted on loop 1 by loop 2, while F21 is the force exerted on loop 2 by loop 1.

This law is not just a theoretical concept; it plays a crucial role in understanding electromagnetism. By proving that the magnetic forces between the loops follow this law, we can better grasp the symmetry and behavior of magnetic fields. This demonstration involves calculus to sum (integrate) the forces over the loop paths, showing they act in opposite directions.
Biot-Savart Law
The Biot-Savart Law is a cornerstone of magnetostatics, derived from Ampère's circuital law and Coulomb's law. It allows us to calculate the magnetic field induced by a steady current in a wire. The formula is expressed as:dB=μ04πIdr×r^r2- Here, - dB is the infinitesimal magnetic field produced by a current element - μ0 is the permeability of free space - I is the current - dr is the current element length vector
- r^ is the unit vector from a point on the wire to the point where the field is measured - and r is the distance between these points.

When applying the Biot-Savart Law to find the forces between current loops, we are interested in the interaction of differential elements, dr1 and dr2. The calculation involves taking into account both the geometry of the loops and applying the cross product to account for directionality, which is vital for proving Newton’s third law through force interactions.
Vector Calculus
Vector Calculus is an essential mathematical tool for understanding and solving problems in the physical sciences, particularly electromagnetism. It involves operations like differentiation and integration applied to vector fields, allowing us to analyze physical quantities varying in space.
In the derivation of forces between electric loops using the Biot-Savart Law, vector calculus comes into play through the manipulation of vector products. When dealing with expressions like dr1×(dr2×s^), we use the vector triple product identity:(a×(b×c))=(ac)b(ab)cThis identity simplifies calculations by converting complex cross products into easier-to-manage dot products and simpler vectors.

Through integration, vector calculus allows us to both sum infinitesimal forces to capture the total force on a loop and to ensure forces meet conditions like Newton’s Third Law. This enables the precise analysis necessary in electromagnetism, from simple circuits to complex magnetic fields.

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Most popular questions from this chapter

In Section 1.5 we proved that Newton's third law implies the conservation of momentum. Prove the converse, that if the law of conservation of momentum applies to every possible group of particles, then the interparticle forces must obey the third law. [Hint: However many particles your system contains, you can focus your attention on just two of them. (Call them 1 and 2.) The law of conservation of momentum says that if there are no external forces on this pair of particles, then their total momentum must be constant. Use this to prove that F12=F21.]

One of the many uses of the scalar product is to find the angle between two given vectors. Find the angle between the vectors b=(1,2,4) and c=(4,2,1) by evaluating their scalar product.

Conservation laws, such as conservation of momentum, often give a surprising amount of information about the possible outcome of an experiment. Here is perhaps the simplest example: Two objects of masses m1 and m2 are subject to no external forces. Object 1 is traveling with velocity v when it collides with the stationary object 2. The two objects stick together and move off with common velocity v. Use conservation of momentum to find v in terms of v,m1, and m2

Prove that if v(t) is any vector that depends on time (for example the velocity of a moving particle) but which has constant magnitude, then v˙(t) is orthogonal to v(t). Prove the converse that if v˙(t) is orthogonal to v(t), then |v(t)| is constant. [Hint: Consider the derivative of v2.] This is a very handy result. It explains why, in two- dimensional polars, dr^/dt has to be in the direction of ϕ^ and vice versa. It also shows that the speed of a charged particle in a magnetic field is constant, since the acceleration is perpendicular to the velocity.

Given the two vectors b=x^+y^ and c=x^+z^ find b+c,5b+2c,bc, and b×c

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