Consider an infinitely long solid metallic cylinder having axis along kˆ . Consider a plane passing through axis of cylinder cutting it in two equal parts. In one part is a uniformly distributed current I1kˆ and in another part is a uniformly distributed current −I2kˆ. As always, task is simple, find the magnitude of magnetic field on the axis of cylinder in μT.
To find the magnitude of the magnetic field on the axis of the cylinder, we can use the Biot-Savart law. The law states that the magnetic field at a point due to a current element is directly proportional to the magnitude of the current and inversely proportional to the square of the distance between the point and the current element.
In this scenario, we have a uniformly distributed current in two equal parts of the cylinder, with currents I1 and -I2, respectively. Let's assume the radius of the cylinder is 'a', and we want to find the magnetic field on the axis of the cylinder.
To apply the Biot-Savart law, we need to integrate the contributions from each current element along the axis of the cylinder. Since the current is uniformly distributed, we can assume a small segment of length 'dz' along the axis of the cylinder.
The magnetic field at a point on the axis due to one of the current elements can be calculated using the Biot-Savart law:
dB = (μ0 / 4π) * (I * (dL x r)) / r^3
Where:
- dB is the magnetic field at a point on the axis
- μ0 is the permeability of free space (4π x 10^-7 T m/A)
- I is the current in the current element
- dL is the differential length element along the current element
- r is the distance between the current element and the point on the axis
- x denotes the cross product
Since we have a symmetric setup with opposite currents I1 and -I2 on either side of the plane, we can integrate from the axis of the cylinder to either side and multiply the result by 2 to get the total magnetic field.
To calculate the magnetic field at the point on the axis due to each current element, we can use the right-hand rule to determine the direction of the cross product. The cross product of the differential length element (dL) with the vector 'r' gives the direction of the magnetic field. By symmetry, we can see that the magnetic fields from the two sides will have opposite directions and will cancel each other out on the axis.
Therefore, the magnitude of the magnetic field on the axis of the cylinder due to this current distribution is zero.
In summary, the magnitude of the magnetic field on the axis of the cylinder is zero μT.