(a) Write an expression of magnetic moment associated with a current (I) carrying circular coil of radius r having N turns.
(b) 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).
OR
(a) Define current sensitivity of a galvanometer. Write its expression.
(b) A galvanometer has resistance G and shows full scale deflection for current Ig .
(i) How can it be converted into an ammeter to measure current up to I0 (I0 > Ig ) ?
(ii) What is the effective resistance of this ammeter ?

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Detailed Solution

(a) magnetic moment $= \begin{aligned} M = NIA \\ M = {NIπr}^{2} \end{aligned}$
According to Biot-sevart law, $\vec{dB} = \frac{μoI}{4 π} \frac{| dl \times \vec{r} |}{r^{3}}$
${dB}^{\to} = \frac{μoI}{4 π} \frac{dl}{r^{2}}$
${dB}_{⊥}$ components due to diametrically opposite components cancel out. Only 𝑑𝐵𝑥 components refrain
$\begin{array}{l} {dB}_{x} = \frac{μ_{0} Idl}{4 {πr}^{2}} \cdot \cos θ \\ B = \int {dB}_{x} \\ B = \frac{μ_{0} {IR}^{2}}{2 {(R^{2} + x^{2})}^{3 / 2}} (along x axis) \end{array}$
OR
a) Deflection per unit current
$I_{s} = \frac{θ}{I} = \frac{BNA}{K}$
b) (i) By connecting a low resistance (R_s) in parallel to galvanometer such that
$(I_{0} - I_{g}) R_{s} = I_{g} G$
(ii) effective resistance
$\begin{array}{l} \frac{1}{R_{A}} = \frac{1}{R_{S}} + \frac{1}{G} = \frac{G + R_{S}}{R_{S} G} \\ ∴ R_{A} = \frac{R_{S} G}{G + R_{S}} \end{array}$