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Question: A long straight cable contains \[7\] strands of wire each carrying a current of \[5A\]. The magnetic...

A long straight cable contains 77 strands of wire each carrying a current of 5A5A. The magnetic induction at a distance of 2cm2cm from the axis of the cable is:
A) 5×10−5T5 \times {10^{ - 5}}T
B) 3.5×10−6T3.5 \times {10^{ - 6}}T
C) 3.5×10−4T3.5 \times {10^{ - 4}}T
D) 1.1×10−3T1.1 \times {10^{ - 3}}T

Explanation

Solution

We know that the current given in the question indicates the amount of current carried by a single strand but the number of strands here is seven. So we can find the total current. Also, the magnetic induction implies the magnetic field around the straight cable.

Complete step by step answer:
We know that Biot Savart law gives us the magnetic field around a small current-carrying conductor.
Let the concerned conductor be of finite length, proceeding with the Biot Savart law that states:
Magnetic field around a small current-carrying conductor at a point is directly proportional to the dot product of current enclosed by the conductor and the length of the conductor and is inversely proportional to the square of the distance between the conductor and the given point.
In the case of a conductor of finite length, we obtain:
B=μoIenclosed2πrB = \dfrac{{{\mu _o}{I_{enclosed}}}}{{2\pi r}}
Where,
Ienclosed={I_{enclosed}} = Total Current enclosed by the conductor
B=B = Magnetic field or magnetic induction
r=r = Distance between the conductor and the point
μo={\mu _o} = Permeability of the medium
=4π×10−7T= 4\pi \times {10^{ - 7}}T
Now, here the current of 5A5A flows through a single strand, therefore, total current through 77 strands is =7×5=35A = 7 \times 5 = 35A
Substituting the values of all the variables in the equation:
B=4π×10−7×352×3.14×2×10−2B = \dfrac{{4\pi \times {{10}^{ - 7}} \times 35}}{{2 \times 3.14 \times 2 \times {{10}^{^{ - 2}}}}}
Thus, we obtain:
B=3.5×10−4TB = 3.5 \times {10^{ - 4}}T
This is the required solution.

Thus, option (C) is correct.

Note: Biot Savart Law and hence the formula can only be applied for steady currents only. The direction of magnetic induction is perpendicular to the plane containing the conductor and the point where the magnetic field is to be calculated. The current enclosed by the conductor is the total current that the conductor carries.