Question

A toroidal core with 3000 turns has inner and outer radii of 11 cm and 12 cm respectively, when a current 0.70 A is passed, the magnetic field produced in the core is 2.5T. Find the relative permeability of the core. (\[{{\mu }_{0}}=4\pi \times {{10}^{-7}}Tm/A\])

Answer 1

Hint: The relative permeability of the core in a toroid or a solenoid is directly proportional to its magnetic field. We can use the simple relation from Ampere’s circuital law to find the magnetic field to find the permeability of the core easily.

Complete step-by-step solution:
We know that toroid is a circular solenoid, which like the solenoid produces a magnetic field when a current is flowing through it. We know from Ampere’s circuital law that the magnetic field passing through a surface due to a current-carrying circuit is given as –
\[\oint{B.dl}=\mu NI\]
Where, \[\mu \]is the permeability of the medium,
N is the number of turns in a coil,
‘I’ is the current through the coil,
‘dl’ is the surface element.
For a toroid this relation becomes –
\[B=\dfrac{\mu NI}{2\pi r}\]
Where r is the mean radius of the toroid.
We can find the required solution using this formula. The unknown quantity is the permeability of the medium inside the toroid core.
So, we can rewrite the given relation to get the permeability as –
\[\begin{align}
  & B=\dfrac{\mu NI}{2\pi r} \\
 & \Rightarrow \mu =\dfrac{B(2\pi r)}{NI} \\
 & \text{given,} \\
 & B=2.5T, \\
 & r=\dfrac{11+12}{2}=11.5cm=0.115m, \\
 & N=3000turns, \\
 & I=0.70A \\
\end{align}\]
Now, we can find the permeability by substituting the above information as –
\[\begin{align}
  & \mu =\dfrac{B(2\pi r)}{NI} \\
 & \Rightarrow \mu =\dfrac{2.5(2\pi \times 0.115)}{3000(0.70)} \\
 & \Rightarrow \mu =8.6\times {{10}^{-4}}Tm/A \\
\end{align}\]
We need the relative permeability of the medium, which is given as the ratio between the permeability of the medium and the permeability of air as –
\[\begin{align}
  & {{\mu }_{r}}=\dfrac{\mu }{{{\mu }_{0}}} \\
 & \text{given,} \\
 & {{\mu }_{0}}=4\pi \times {{10}^{-7}}Tm/A \\
 & \Rightarrow {{\mu }_{r}}=\dfrac{8.6\times {{10}^{-4}}}{4\pi \times {{10}^{-7}}} \\
 & \therefore {{\mu }_{r}}=684.37 \\
\end{align}\]
So, we get the relative permeability of the core of the toroid as 684.37.

Note: The relative permeability of a medium is the measure of how better the medium is in producing a magnetic field around a coil than the usual air medium. IN most cases, we use other mediums instead of the air to get better results from the same setup.