Question

Calculate magnetic flux density of the magnetic field at the center of a circular coil of 50 turns, having radius of 0.5m and carrying a current of 5A.

Answer 1

Hint- In this question we will use the concept of Biot-Savart law which states that the magnetic field intensity is directly proportional to the length of coil, inversely proportional to the square of distance between the element and the point, directly proportional to the current etc. The formula is given below

$d\overrightarrow B = \dfrac{{{\mu _0}i\overrightarrow {dl} \times \overrightarrow r }}{{4\pi {r^3}}}$

Complete step by step answer:
Now the total magnetic field due to coil can be calculated by integrating from 0 to $2\pi $

\[\begin{gathered}

B = \int\limits_0^{2\pi } {dB} = \dfrac{{{\mu _0}}}{{4\pi }} \times \dfrac{1}{{{a^2}}}\int\limits_0^{2\pi } {dl} \\
   \Rightarrow B = \dfrac{{{\mu _0}}}{{4\pi }} \times \dfrac{1}{{{a^2}}} \times 2\pi a \\
   \Rightarrow B = \dfrac{{{\mu _0}I}}{{2a}} \\

\end{gathered} \]

When coil has n turns then

\[ \Rightarrow B = \dfrac{{{\mu _0}IN}}{{2a}}\]

Where B is the magnetic field due to coil
I is the current , a is the radius of the coil
N is the number of turns.
Now we will use the above formula to solve the question

Given

I = 5A
N = 50
a = 0.5m

Substituting these values in the formula derived above for a circular coil, we get

\[
   \Rightarrow B = \dfrac{{{\mu _0}IN}}{{2a}} \\
   \Rightarrow B = \dfrac{{5 \times 50}}{{2 \times 0.5}}{\mu _0} \\
   \Rightarrow B = 250{\mu _0} T \\
\]

Hence, magnetic field at the center of the coil is \[250{\mu _0} T\]

Additional information- The magnetic field at the center of a circular coil carrying current is double if the current flowing through the coil is doubled and the radius of the coil is halved. This is because if current is doubled the magnetic field is also double because it is directly proportional to the current and inversely proportional to the radius.

Note- In order to solve these types of questions, remember the biot- savart's law. The formula derived above using the biot- savart law for a circular coil can be used for a semi-circular coil, semicircular arc and many more. The biot- savart law can be used to find the magnetic field of any regular shape by integrating along the path same as we derived for the circular coil.