Chapter 36, Problem 21A

The electromagnets A, B, and C are shown below. Note the number of turns of wire and the current in each. The iron cores are Identical. Rank the electromagnets in magnetic field intensity at point P (just outside each electromagnet), from strongest to weakest.

To determine

To find: The order of magnetic field intensity at a given point from strongest to weakest on the bases of the number of turns

Answer to Problem 21A

The order of the field’s intensity is

  $\mathrm{Electromagnet}B>ElectromagnetC>ElectromagnetA$

Electromagnet B has the strongest field’s intensity while electromagnet A has the lowest field’s intensity.

Explanation of Solution

Given info:

1. The number of turns of electromagnet A, B, and C are $5,5,and3$respectively.
2. The current flows through the electromagnet A, B and C are $2ampere,4ampereand4ampere$ respectively.

Formula used:

When the coil has N number of turns then the magnetic field along its axis is calculated as,

  $\mathrm{B}=\frac{{\mu }_{°}}{2}\frac{IN{a}^2}{{\left(x^2+a^2\right)}^{3/2}}Tesla$

Here, B is the magnetic field’s strength

“N” is the number of turns in the coil

“I” is the current that flows through the electromagnet

“a” is the area of cross section of the wire

Calculation:

Case 1: For electromagnet A

  $\begin{array}{l} N=5\hfill \\ I=2ampere\hfill \end{array}$

The field’s strength is

  $\begin{array}{l}B=2\times 5\left(\frac{{\mu }_{°}}{2}\frac{a^2}{{\left(x^2+a^2\right)}^{3/2}}\right)Tesla\\ B_A=10\left(\frac{{\mu }_{°}}{2}\frac{a^2}{{\left(x^2+a^2\right)}^{3/2}}\right)Tesla\end{array}$

Similarly, for electromagnet B

  $N=5andI=4ampere$

  $\begin{array}{l}B=4\times 5\left(\frac{{\mu }_{°}}{2}\frac{a^2}{{\left(x^2+a^2\right)}^{3/2}}\right)Tesla\\ B_B=20\left(\frac{{\mu }_{°}}{2}\frac{a^2}{{\left(x^2+a^2\right)}^{3/2}}\right)Tesla\end{array}$

For electromagnet C:

  $N=3andI=4ampere$

  $\begin{array}{l}B=3\times 4\left(\frac{{\mu }_{°}}{2}\frac{a^2}{{\left(x^2+a^2\right)}^{3/2}}\right)Tesla\\ B_C=12\left(\frac{{\mu }_{°}}{2}\frac{a^2}{{\left(x^2+a^2\right)}^{3/2}}\right)Tesla\end{array}$

Conclusion:

Hence, the order of the field’s intensity is

  $\mathrm{Electromagnet}B>ElectromagnetC>ElectromagnetA$