Question

If you have some experience in electromagnetism and with vector calculus, prove that the magnetic forces, ${\mathbf{F}}_{12}$ and ${\mathbf{F}}_{21}$, between two steady current loops obey Newton's third law. [Hints: Let the two currents be $I_1$ and $I_2$ and let typical points on the two loops be ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$. If $d{\mathbf{r}}_1$ and $d{\mathbf{r}}_2$ are short segments of the loops, then according to the Biot- Savart law, the force on $d{\mathbf{r}}_1$ due to $d{\mathbf{r}}_2$ is $$\frac{{\mu }_0}{4\pi }\frac{I_1{I}_2}{s^2}d{\mathbf{r}}_1\times (d{\mathbf{r}}_2\times \hat{\mathbf{s}})$$ where $\mathbf{s}={\mathbf{r}}_1-{\mathbf{r}}_2.$ The force ${\mathbf{F}}_{12}$ is found by integrating this around both loops. You will need to use the " $BAC-CAB$ " rule to simplify the triple product.]

Short answer

The forces ${\mathbf{F}}_{12}$ and ${\mathbf{F}}_{21}$ are equal and opposite, satisfying Newton's third law.

Step-by-step solution

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.

Biot-Savart Law Application

The force on a segment of loop 1 due to a segment on loop 2 is given by $\frac{{\mu }_0}{4\pi }\frac{I_1{I}_2}{s^2}d{\mathbf{r}}_1\times (d{\mathbf{r}}_2\times \hat{\mathbf{s}})$, where $\mathbf{s}={\mathbf{r}}_1-{\mathbf{r}}_2$. This is derived from the Biot-Savart law.

Simplify the Triple Product

Using the vector identity $(\mathbf{a}\times (\mathbf{b}\times \mathbf{c}))=(\mathbf{a}\cdot \mathbf{c})\mathbf{b}-(\mathbf{a}\cdot \mathbf{b})\mathbf{c}$, simplify $d{\mathbf{r}}_1\times (d{\mathbf{r}}_2\times \hat{\mathbf{s}})$.

Express the Forces

The simplified expression for the force becomes $d{\mathbf{F}}_{12}=\frac{{\mu }_0}{4\pi }\frac{I_1{I}_2}{s^2}((d{\mathbf{r}}_1\cdot \hat{\mathbf{s}})d{\mathbf{r}}_2-(d{\mathbf{r}}_1\cdot d{\mathbf{r}}_2)\hat{\mathbf{s}})$.

Integrate Over the Loops

Integrate this expression over both loops to find the total forces ${\mathbf{F}}_{12}$ and ${\mathbf{F}}_{21}$.

Prove Newton's Third Law

By symmetry and integration limits, the forces on the segments of each loop will balance out, showing ${\mathbf{F}}_{12}=-{\mathbf{F}}_{21}$, thus proving they satisfy Newton's third law.

Key concepts

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 ${\mathbf{F}}_{12}=-{\mathbf{F}}_{21}$. Here, ${\mathbf{F}}_{12}$ is the force exerted on loop 1 by loop 2, while ${\mathbf{F}}_{21}$ 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:$$d\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{Id\mathbf{r}\times \hat{\mathbf{r}}}{r^2}$$- Here, - $d\mathbf{B}$ is the infinitesimal magnetic field produced by a current element - ${\mu }_0$ is the permeability of free space - $I$ is the current - $d\mathbf{r}$ is the current element length vector
- $\hat{\mathbf{r}}$ 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, $d{\mathbf{r}}_1$ and $d{\mathbf{r}}_2$. 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 $d{\mathbf{r}}_1\times (d{\mathbf{r}}_2\times \hat{\mathbf{s}})$, we use the vector triple product identity:$$(\mathbf{a}\times (\mathbf{b}\times \mathbf{c}))=(\mathbf{a}\cdot \mathbf{c})\mathbf{b}-(\mathbf{a}\cdot \mathbf{b})\mathbf{c}$$This 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.