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: $$\frac{F}{l} = \frac{\mu_{0} I^2}{2 \pi d} \mathrm{N} / \mathrm{m}$$