Question

Two conducting strips, having infinite length in the \(z\) direction, lie in the \(x z\) plane. One occupies the region \(d / 2

Short answer

(b) As the width of one of the strips approaches zero while maintaining constant current, what is the force per unit length?

Step-by-step solution

Determine the magnetic field of the two strips

To find the magnetic field of the two strips, we can use the Biot-Savart law which states that the magnetic field produced by an electric current is given by: $$\mathrm{d} \mathbf{B}=\frac{\mu_{0}}{4 \pi} \frac{I \mathrm{d} \mathbf{l} \times \mathbf{r}}{\left|\mathbf{r}\right|^{3}}$$ Since the strips are infinite, we can simplify this calculation by using the Ampère's Circuital law: $$\oint \mathbf{B} \cdot d \mathbf{l}=\mu_{0} I$$ We consider a rectangular path between the strips and parallel to them, passing through each strip. On both ends, magnetic fields are perpendicular to the path, so the integral along those sections cancel out. The integrals on the other two sections are equal and opposite, so they add up. We can solve for B: $$2Bl = \mu_{0} K_0 b$$ $$\mathbf{B} = \frac{\mu_{0} K_0 b}{2l} \mathbf{a}_{y}$$ This must be evaluated separately for both strips in different positions.

Evaluate the magnetic force on each strip

The magnetic force on each strip is given by the Lorentz force: $$\mathbf{F} = I (\mathbf{l} \times \mathbf{B})$$ Using the magnetic field expression, for a unit length in the \(z\) direction, we have: $$\mathbf{F} = \pm K_{0} b (\frac{\mu_{0} K_{0} b}{2l} \mathbf{a}_{y})$$ For the positive strip, the force is inward, and for the negative strip, the force is outward.

Calculate the force per unit length that tends to separate the two strips

The force per unit length can be found by dividing the force by the length of the strip in the \(z\) direction. The force per unit length on the positive strip is given by: $$\frac{F}{l} = -\frac{\mu_{0} K_{0}^2 b^2}{4}$$ Similarly, the force per unit length on the negative strip is given by: $$\frac{F}{l} = \frac{\mu_{0} K_{0}^2 b^2}{4}$$ The total force per unit length tends to separate the two strips.

Analyze the force per unit length as the width of one of the strips approaches zero

,We are given that as \(b\) approaches zero, the current remains constant: \(I = K_0 b\). Therefore, we rewrite the force per unit length as: $$\frac{F}{l} = \pm \frac{\mu_{0} I^2}{2 \pi d}$$ As the width \(b\) of one of the strips approaches zero, the force per unit length tends to: $$\frac{F}{l} = \frac{\mu_{0} I^2}{2 \pi d} \mathrm{N} / \mathrm{m}$$

Key concepts

magnetic field calculation

Magnetic fields are regions where a force is exerted on magnets or moving charged particles. Calculating the magnetic field generated by current involves understanding how currents interact with space around them. In the case of two infinite conducting strips, each carries a surface current density, and they create magnetic fields that affect each other.

The Biot-Savart law provides a way to calculate the magnetic field generated by a point current. But for infinitely long currents, Ampère's Circuital Law is more practical because it simplifies the calculations using symmetry. This law works by considering a closed loop surrounding the current and relates the magnetic field along the loop to the current passing through it.

For our strips, we use a path that surrounds both strips but only interacts perpendicularly at parts of the rectangle around the currents. The two contributing sections of the path parallel to these currents give rise to the total field. In this exercise, by using Ampère's law, the magnetic field of the strip can be expressed as: \[\mathbf{B} = \frac{\mu_{0} K_0 b}{2l} \mathbf{a}_{y}\] which results from simplifying the effects of infinitely long strips on a symmetric path.

surface current density

Surface current density, represented as \( \mathbf{K} \), is a measure of how much current flows through a unit width of a surface. It is distinct from bulk current density as it not only considers how much current passes through a surface but does so in a specified direction and per unit width along the surface. This is essential in cases where currents are constrained to flow on a surface, like in thin conducting strips.

In our scenario, the strips have surface currents flowing in opposite directions: one with a density \( \mathbf{K}=K_{0} \mathbf{a}_{z} \) and the other \( -K_{0} \mathbf{a}_{z} \).

The differentiation in surface current density essentially describes how the currents on each strip not only contribute to the magnetic fields but also how these fields interact with each other. Oppositely flowing currents imply there is a tendency for these fields to repel each other, directly leading to forces trying to separate the strips.

Biot-Savart law

The Biot-Savart Law is a fundamental equation used for calculating the magnetic field produced at a point in space by a current-carrying element. It is expressed as:\[\mathrm{d} \mathbf{B} = \frac{\mu_{0}}{4 \pi} \frac{I \mathrm{d} \mathbf{l} \times \mathbf{r}}{\left|\mathbf{r}\right|^{3}}\]This equation implies that every small current element produces a magnetic field at a point in space, depending inversely on the cube of the distance. It effectively generalizes how current modifies space through magnetic effects.

While this law is perfect for point currents, when dealing with infinitely long conductors or large setups, simplifying it using laws like Ampère's Circuital Law is often more efficient.

In our exercise, though the Biot-Savart law introduces the problem, we sidestep the need to integrate across these infinitely large areas directly, by applying the much simpler Ampère's Circuital Law to get a solution in terms of magnetic field.

Ampère's Circuital law

Ampère's Circuital Law is one of the cornerstones for calculating magnetic fields in continuous current distributions. Simplified as \[\oint \mathbf{B} \cdot d \mathbf{l} = \mu_{0} I\]this law relates the integrated magnetic field around a closed loop to the current passing through that loop. It hugely simplifies calculations of magnetic fields for systems with high symmetry, like infinite lines or surfaces of current.

In the exercise, this law allows for the calculation of the fields surrounding our two strips, by considering a loop that neatly encompasses the surface currents. Segments where the path is not perpendicular to the surface currents cancel out, allowing for a straightforward calculation of field strength.

Thus, Ampère's Circuital Law provides a practical framework to understand interactions of configurations like these strips, especially when the goal is to determine forces resulting from their electromagnetic interactions. It explains how electromagnetic forces, relying on the direction and magnitude of surface currents, create tendencies for the strips to move or orient in space, driven purely by their fields.