Question

Two semi-infinite filaments on the \(z\) axis lie in the regions \(-\infty

Short answer

Question: Given an arrangement of two semi-infinite filaments carrying current I, find the magnetic field \(\mathbf{H}\) at a point \((\rho, \phi, z=0)\). Calculate the value of \(a\) for which the magnitude of \(\mathbf{H}\) becomes half the value for an infinite filament. Solution: This problem is solved using the Biot-Savart law to find the individual contributions to the magnetic field, \(\mathbf{H}\), due to each filament. The total magnetic field is calculated by adding the contributions. Finally, the value of \(a\) that causes the magnitude to be half the value for an infinite filament is found. Steps for solving the problem include defining the magnetic field due to a single filament using Biot-Savart law, computing the magnetic field due to each filament, calculating the total magnetic field, and finding the value of \(a\) that reduces the magnitude by half.

Step-by-step solution

Define magnetic field due to a single filament using Biot-Savart law

The magnetic field \(\mathbf{dB}\) due to a small current element \(I \, d\mathbf{l}\) at point \(P\) with position vector \(\mathbf{r}\) is given by Biot-Savart law: $$ \mathbf{dB}=\frac{\mu_0}{4\pi}\frac{I \, d\mathbf{l} \times \mathbf{e}_r}{r^2} $$ In cylindrical coordinates \((\rho, \phi, z)\), we have \(d\mathbf{l}=dz'\mathbf{a}_z\), \(\mathbf{e}_r=-\mathbf{a}_\rho\) and \(r^2=(\rho^2+z'^2)\). Remember that \(d\mathbf{l}\) is pointed in the \(z\) direction.

Compute the magnetic field due to each filament

For each semi-infinite filament: the magnetic field contributions \(\mathbf{dB}\) will have \(\rho\) and \(\phi\) components and we need to compute these components separately. Thus, we have: $$ \mathbf{dB}_\rho=-\frac{\mu_0 I}{4\pi}\frac{dz'}{\rho^2+z'^2} \sin\phi $$ and $$ \mathbf{dB}_\phi=-\frac{\mu_0 I}{4\pi}\frac{dz'}{\rho^2+z'^2} \cos\phi $$ Now, compute the magnetic field due to each filament by integrating over their entire lengths, assuming the filament is located at \(z'=a\) and \(z'=-a\) respectively, i.e., integrate over \(-\infty<z'<-a\) and \(a<z'<\infty\).

Calculate the total magnetic field

The total magnetic field \(\mathbf{H}(\rho,\phi)\) is the sum of the magnetic fields due to both filaments. After adding the contributions, the components \(\mathbf{H_\rho}\) and \(\mathbf{H_\phi}\) can be expressed as a function of \(\rho\) and \(\phi\).

Find the value of \(a\) that reduces the magnitude by half

To find the value of \(a\) that causes the magnitude of \(\mathbf{H(\rho=1,z=0)}\) to be half of the value for an infinite filament, compare the total magnetic field magnitudes calculated in the previous step. Solve for \(a\) that makes their ratio equal to \(1/2\).