Question

A straight vertical conductor carries a current. At a point \[5cm\] due north of it, the magnetic induction is found to be \[20T\] due east. The magnitude of magnetic induction at a point \[10cm\] east of it will be
A. \[20T\]
B. \[40T\]
C. \[5T\]
D. \[10T\]

Answer 1

Hint: The magnetic induction is inversely proportional to the distance. If the magnetic induction is high, the distance is less. The quantity of magnetic induction and the distance are inversely proportional.

Formula used:
\[B\alpha \dfrac{1}{r}\]
To remove the proportionality constant, it gives an equation. This expression gives the magnitude of the magnetic field around the wire. The final equation is,
\[B = \dfrac{{{\mu _0}I}}{{2\pi r}}\]
Where,
\[{\mu _0}\]- Permeability of free space, it has a constant value, \[{\mu _0} = 4\pi \times {10^{ - 7}}Tm/A\]
\[B\] - Magnetic induction
\[r\] – distance between the conduction and measured point

Complete step by step solution:
\[{B_1}\] is the magnetic induction produced at the distance \[5cm\] of north, here \[{B_1} = 20T\], \[{r_1} = 5cm\] and \[{B_2}\] is the magnetic induction produced at the distance \[10cm\] of east, \[{r_2} = 10cm\].
\[{B_1} = \dfrac{1}{{{r_1}}}\]
\[{B_2} = \dfrac{1}{{{r_2}}}\]
\[\dfrac{{{B_1}}}{{{B_2}}} = \dfrac{{{r_2}}}{{{r_1}}}\]
Substitute the values in the above equation,
\[\dfrac{{20T}}{{{B_2}}} = \dfrac{{10}}{5}\]
\[{B_2} = 10T\]
Hence the correct option is D.

Additional Information:
Magnetic induction was first discovered by Michael Faraday and published in \[1831\] . After that \[1831\] faraday shows an experimental demonstration to prove magnetic induction. Faraday explained the phenomenon of electromagnetic induction using the concept of lines of force. But all the scientists rejected his ideas at that time because they were not expressed and explained mathematically. But one scientist named James Clerk Maxwell used Faraday's ideas as the basis of his electromagnetic theory.

Note:
If the wire is not straight, or it’s a circular loop, the magnetic induction around the circular loop has a pattern similar to the bar magnet field lines. The loop has a radius \[R\], the equation for the magnetic induction in a circular loop is,
\[B = \dfrac{{{\mu _0}I}}{{2R}}\]
If the loop consists of a \[N\] number of turns, then the equation for the magnetic induction will be written as,
\[B = N\dfrac{{{\mu _0}I}}{{2R}}\]