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Moving Charges and Magnetism — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsMoving Charges and Magnetism3 Marks
Question
  1. Write an expression of magnetic moment associated with a current (I) carrying circular coil of radius r having N turns.
  2. Consider the above mentioned coil placed in YZ plane with its centre at the origin. Derive expression for the value of magnetic field due to it at point (x, 0, 0).
✓

Answer

  1. Magnetic moment $\vec{\text{M}}=\text{Ni}(\pi\text{r}^2)\hat{\text{n}}.$
  2.  

Magnetic field at point P(x, 0, 0) due to $\vec{\text{Idl}}$
$\vec{\text{dB}}=\frac{\mu_0}{4\pi}\frac{\text{Idl}\sin90^\circ}{\text{r}^2_1}\text{along PQ}$
For entire coil $\int\vec{\text{dB}}\cos\theta=0$
$\therefore\vec{\text{B}}\text{ at }\text{P}$
$\Rightarrow\text{B}=\int\text{dB}\sin\theta=\frac{\mu_0\text{I}\sin\text{R}}{4\pi\text{R}^2}\int\limits^{2\pi\text{r}}_{0}\text{dl}$
$=\frac{\mu_0\text{I}}{4\pi\text{r}^2_1}\times\frac{\text{r}}{\text{r}_1}\times(2\pi\text{r})$
$\Rightarrow\vec{\text{B}}=\frac{\mu_0\text{Ir}^2}{2\big(\text{r}^2+\text{x}^2\big)^\frac{3}{2}}\hat{\text{i}}$
Coil has N turns then,
$\vec{\text{B}}=\frac{\mu_0\text{INr}^2}{2\big(\text{r}^2+\text{x}^2\big)^\frac{3}{2}}\hat{\text{i}}$

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