Question

magnetic field at any point which is at a distance R due to a long straight conductor carrying current varies as ?​

Answer 1

Answer:

Magnetic field at a distance r from an infinitely long straight conductor carrying a steady current varies as. 1) I/r^2. 2) 1/r.

Answer 2

Given :

Long curent carrying condctor.

To find :

Magnetic field at point at a distance R.

Solution :

According to Biot-savart law

$dB = \frac{ \mu_{0}Idl\times r}{4\pi R^3} \:$

(equation 1)

Integrating this equation we get

$B = \int \frac{ \mu_{0}Idl\times r}{4\pi R^3} \: \\ B = \int \frac{ \mu_{0}Idy\sin \: \theta}{4\pi R^2} \:$

Putting

y = R tanθ

dy = Rsec²θ

and Integrating we get,

Magnetic field at point distance R perpendicular to wire :

$B = \frac{ \mu_{0}I(\sin \theta_{1} +\sin \theta_{2} )}{4\pi R} \: \:$

For very long wires

$\theta_{1} = \theta_{2} = 90 \degree \\ so \\ B = \frac{ \mu_{0}I(2\sin 90 )}{4\pi R} \: \: \\ so \\ B = \frac{ \mu_{0}I}{2\pi R}$

As we can see that

Magnetic field B is inversely proportional to R

So magnetic field at R varies as

$B \propto \frac{1} {R}$