Question

At a distance of 1 cm from a very long straight wire carrying a current of 5 A, Calculate the flux density. At what distance from the wire will the field flux density neutralize that due to the earth’s horizontal component flux density $2 \times {10^{ - 5}}$ T ( ${\mu _o} = 4\pi \times {10^{ - 7}}H{m^{ - 1}}$ )

Answer 1

Hint: Here we are going to use two concepts related to magnetic field
1) Remember magnetic field at a point and flux density at a point both are similar terms that means there is no difference between magnetic field and flux density.
The formula we are going to use for calculating flux density of infinite wire is $\dfrac{{{\mu _o}I}}{{2\pi a}}$ where ${\mu _o}$ permeability constant, a is the perpendicular distance from the wire and $I$ is the current flowing in the infinite wire.
2) Secondly, the point where the net magnetic field becomes zero is called the neutral point. In simple words at a neutral point, earth’s magnetic field is equal and opposite to the applied magnetic field.

Complete step by step answer:
According to the question given, we will divide it into two parts and in both the parts we will going to use the formula of magnetic field of infinite wire $\dfrac{{{\mu _o}I}}{{2\pi a}}$
1) In the given that:
a = 1cm = $1 \times {10^{ - 2}}m$
$I = 5A$
So, let us solve it by using the formula discussed above,
B= $\dfrac{{{\mu _o}I}}{{2\pi a}}$
Putting the values of current and distance from the wire ‘a’,
$\implies$ B$ = \dfrac{{4\pi \times {{10}^{ - 7}} \times 5}}{{2\pi \times 1 \times {{10}^{ - 2}}}}$
$\implies$ B $ = {10^{ - 4}}T$

2) Earth’s magnetic field is given which is equal to $2 \times {10^{ - 5}}T$
Now this field will be equal to the external field that is field by the infinite wire say at a distance of $x$
Again, applying the same formula and equating,
$\dfrac{{{\mu _o}I}}{{2\pi x}} = 2 \times {10^{ - 5}}$
$
= \dfrac{{4\pi \times {{10}^{ - 7}} \times 5}}{{2\pi \times x \times {{10}^{ - 2}}}} = 2 \times {10^{ - 5}} \\
x \times {10^{ - 2}} = 5 \\
$
Hence x = 5 cm so the neutral point will be at 5 cm.

Note: In the further part, many questions can ask about the direction in which the neutral point is coming, so it will be due east because in the east direction, Flux density lines due to wire will be running against the magnetic field due to earth so both will cancel each other.