physics-class-12-unit-04-chapter-07-moving-coil-galvanometer-ammeter-and-voltmeter-l-7-7-tzsnbfapwwk

<h3 id="success-stylefont-size25px-moving-coil-galvanometer-ammeter-and-voltmeter-l-7-success-1"><Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success></h3>
<h3 id="primary-stylefont-size24px-torque-primary"><primary style="font-size:24px"> Torque </primary></h3>
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<p>$\vec{\tau}=\vec{m}\times \vec{B}$</p>
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<p>$\vec{m}:$ Magnitude dipole moment</p>
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<p>$\vec{B}:$ Magnetic field</p>
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<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/23fb1319-eac5-4be9-bf6c-c0d4d7b3ca00/public"  width="100%" >

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<footer style = "font-size: 18px"> $\rightarrow$ Moving Coil Galvanometer Ammeter and Voltmeter $\rightarrow$ <span style = "color:blue"> Torque </span> $\rightarrow$ Moving Coil Galvanometer $\rightarrow$ Moving Coil Galvanometer </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Moving Coil Galvanometer </primary>
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- N turns

- A : Area

- $\tau_{current}=\vec{m} \times \vec{B} = mB$

- $= NIAB$

- $\tau_{spring}=k\phi$

- k : Spring constant

- $\phi$ : Angle of rotation

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<footer style = "font-size: 18px">Moving Coil Galvanometer Ammeter and Voltmeter $\rightarrow$ Torque $\rightarrow$ <span style = "color:blue"> Moving Coil Galvanometer </span> $\rightarrow$ Moving Coil Galvanometer $\rightarrow$ Galvanometer as an Ammeter </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Galvanometer as an Ammeter </primary>
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- **Ammeter : Very low resistance**

- $\frac{1}{R}=\frac{1}{R_G}+\frac{1}{R_s}=\frac{R_s + R_G}{R_s R_G}$

- $R=\frac{R_s R_G}{R_s+R_G}$

- $R_s <<< R_G$

- $R \simeq \frac{R_s R_G}{R_G} \simeq R_s$

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<footer style = "font-size: 18px">Moving Coil Galvanometer $\rightarrow$ Moving Coil Galvanometer $\rightarrow$ <span style = "color:blue"> Galvanometer as an Ammeter </span> $\rightarrow$ Example $\rightarrow$ Galvanometer with Coil Resistance </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Galvanometer with Coil Resistance </primary>
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- $I=I_s + I_G$

- $I_c R_G = I_s R_s$

- $I=I_G(1+\frac{I_s}{I_G})=I_G(1+\frac{R_G}{R_S})$

- $I_G.\frac{(R_s + R_G)}{R_S}$

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<footer style = "font-size: 18px">Galvanometer as an Ammeter $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Galvanometer with Coil Resistance </span> $\rightarrow$ Galvanometer with Coil Resistance $\rightarrow$ Galvanometer to a Voltmeter </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Galvanometer to a Voltmeter </primary>
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- $R=R_p + R_G \simeq R_p$

- for $R_p >> R_G$

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<footer style = "font-size: 18px">Galvanometer with Coil Resistance $\rightarrow$ Galvanometer with Coil Resistance $\rightarrow$ <span style = "color:blue"> Galvanometer to a Voltmeter </span> $\rightarrow$ Example $\rightarrow$ Magnetic Field due to a Cicular Loop of Current </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Full seal $I_f = 1mA$

- $R_G = 20 \Omega$

- Voltmeter to measure upto 10V

- $I=I_G + I_c$

- $v = I_G(R_p+R_G)$

- we want $I_G=1mA$ when v=10v

- $R_p = \frac{v-I_G R_G}{I_G}=\frac{10-10^{-3} \times 20}{10^{-3}} \simeq 9980 \Omega$

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<footer style = "font-size: 18px">Galvanometer with Coil Resistance $\rightarrow$ Galvanometer to a Voltmeter $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Magnetic Field due to a Cicular Loop of Current $\rightarrow$ Magnetic Field in the Plane of the Current Loop </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Magnetic Field due to a Cicular Loop of Current </primary>
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- $\vec{B}=\frac{\mu_0 I R^2}{2(2^2+R^2)^{3/2}}\hat{k}$

- For $z >> R$

- $\vec{B} \simeq \frac{\mu_0I R^2}{2z^3}\hat{k}=\frac{\mu_0\hat{m}}{2 \pi 2^3}$

- $|\vec{m}|=I.\pi R^2$

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<footer style = "font-size: 18px">Galvanometer to a Voltmeter $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Magnetic Field due to a Cicular Loop of Current </span> $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field in the Plane of the Current Loop </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Magnetic Field in the Plane of the Current Loop </primary>
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- $x >> R$

- Coordinated of $\vec{dl}$

- $=R \cos \theta, R \sin \theta$

- $\vec{r}=(x-R \cos \theta)\hat{i}-R\sin \theta \hat{j}$

- $\vec{dl}=-dl \sin \theta \hat{i}+dl \cos \theta \hat{j}$

- $=-Rd \theta \sin \theta \hat{i}+Rd \theta \cos \theta \hat{j}$

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<footer style = "font-size: 18px">Example $\rightarrow$ Magnetic Field due to a Cicular Loop of Current $\rightarrow$ <span style = "color:blue"> Magnetic Field in the Plane of the Current Loop </span> $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field in the Plane of the Current Loop </footer>

- $\vec{dl}\times \vec{r}=(-R \sin \theta d \theta \hat{i}+R \cos \theta d \theta \hat{j})\times [(x-R \cos \theta)\hat{i}-R \sin \theta \hat{j}]$

- $=R^2 \sin^2 \theta d \theta \hat{k}+R\cos \theta (x-R \cos \theta)d \theta \hat{k}$

- $=R^2 \sin^2 \theta d \theta \hat{k}-R x \cos \theta d \theta \hat{k}+R^2 \cos^2 d \theta \hat{k}$

- $\vec{dl}\times \vec{r}=R^2 d \theta \hat{k}-Rx \cos \theta d \theta \hat{k}$

- $r^3 = [(x-R \cos \theta)^2+R^2 \sin^2 \theta]^{3/2}$

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-04-chapter-07-moving-coil-galvanometer-ammeter-and-voltmeter-l-7_7-tzsnbfapwwk-10.jpg)

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<footer style = "font-size: 18px">Magnetic Field due to a Cicular Loop of Current $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ <span style = "color:blue"> Magnetic Field in the Plane of the Current Loop </span> $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field in the Plane of the Current Loop </footer>

- $\vec{B}=\frac{\mu_0 I}{4 \pi}\oint \frac{\vec{dl} \times \vec{r}}{r^3} = \frac{\mu_0 I R}{4 \pi} \int_0^{2 \pi}$

- $\frac{(R-x \cos \theta) d \theta}{0[(x-R \cos \theta)^2+R^2 \sin^2 \theta)^{3/2}}\hat{k}$

- $x>>R$

- $(x-R \cos \theta)^2+R^2 \sin^2 \theta = x^2 + R^2 \cos^2 \theta - 2 x R \cos^2 \theta +R^2 \sin^2 \theta$

- $=x^2 + R^2 - 2 x R \cos \theta=x^2[1-\frac{2R}{x}\cos \theta +\frac{R^2}{x^2}]$

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<footer style = "font-size: 18px">Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ <span style = "color:blue"> Magnetic Field in the Plane of the Current Loop </span> $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field in the Plane of the Current Loop </footer>

- $[(x-R \cos \theta)^2+R^2 \sin ^2 \theta]^{3 / 2}$

- $=x^3[1-\frac{2 R}{x} \cos \theta+\frac{R^2}{x^2}]^{3 / 2}$

- $ \frac{1}{[(x-R \cos \theta)^2+R^2 \sin^2 \theta]^{3 / 2}} $

- $ =\frac{1}{x^3}[1-\frac{2 R}{x} \cos \theta+\frac{R^2}{x^2}]^{-3 / 2}$

- $ \simeq \frac{1}{x^3}[1+\frac{3 R}{x} \cos \theta]$

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- $\vec{B}=\frac{\mu_0I R}{4 \pi} \int_0^{2 \pi}(R-x \cos \theta) \frac{1}{x^3}(1+\frac{3 R}{x} \cos) d \theta \hat{k}$

- $ =\frac{\mu_0I R}{4 \pi x^3} \int_0^{2 \pi}(R- 3R \cos^2 \theta - x \cos \theta + \frac{3R^2}{x} \cos \theta) d \theta \hat{k}$

- $=\frac{\mu_0 I R^2}{4 \pi x^3}\int_0^{2 \pi}[1-3(\frac{1+\cos 2 \theta}{2}]d \theta \hat{k}$

- $\cos 2 \theta = 2 \cos^2 \theta -1 $

- $\vec{B} = \frac{\mu_0 I R^2}{4 \pi x^3}\int_0^{2 \pi}(-\frac{1}{2})d \theta \hat{k}$

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- $\vec{B}=-\frac{\mu_0 I R^2 \pi}{4 \pi x^3} \pi \hat{k}$

- $\vec{m}=I \pi R^2 \hat{k}$

- $\vec{B} = -\frac{\mu_0 \vec{m}}{4 \pi x^3}$  $ x>>R$

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<footer style = "font-size: 18px">Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ <span style = "color:blue"> Magnetic Field in the Plane of the Current Loop </span> $\rightarrow$ Magnetic Field of the Current Loop $\rightarrow$ Potential Energy of a Magnetic Dipole Placed in a Magnetic Field </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Magnetic Field of the Current Loop </primary>
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- $\vec{m}$

- $\vec{B}$ (along axis) $= \frac{mu_0 \vec{m}}{2 \pi z^3}$

- $\vec{B}$ (in plane of the loop) $x>>R$ $= -\frac{\mu_0 \vec{m}}{4 \pi x^3}$

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<footer style = "font-size: 18px">Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field in the Plane of the Current Loop $\rightarrow$ <span style = "color:blue"> Magnetic Field of the Current Loop </span> $\rightarrow$ Potential Energy of a Magnetic Dipole Placed in a Magnetic Field $\rightarrow$ External Torque </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Potential Energy of a Magnetic Dipole Placed in a Magnetic Field </primary>
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- $\vec{B}=B\vec{j}$

- $\vec{m}=m \cos \theta \hat{j} + m \sin \theta \hat{k}$

- Torque due to magnetic field

- $\tau_{mag}=\vec{m}\times \vec{B}$

- $=(m \cos \theta \hat{j}+m \sin \theta \hat{k})\times B \hat{k}$

- $=-m B \sin \theta \hat{i}$

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<footer style = "font-size: 18px">Magnetic Field in the Plane of the Current Loop $\rightarrow$ Magnetic Field of the Current Loop $\rightarrow$ <span style = "color:blue"> Potential Energy of a Magnetic Dipole Placed in a Magnetic Field </span> $\rightarrow$ External Torque $\rightarrow$ Zero of Potential Energy </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
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- $\tau_{ext}=m B \sin \theta \hat{i}$

- Change in potential energy in rotating the dipole from $\theta_1$, to $\theta_2$ is

- $U=\int_{\theta_1}^{\theta_2}\tau_{ext} d \theta$

- $=mB \int_{\theta_1}^{\theta_2} \sin \theta d \theta=-mB(\cos \theta_2 - \cos \theta_1)$

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<footer style = "font-size: 18px">Magnetic Field of the Current Loop $\rightarrow$ Potential Energy of a Magnetic Dipole Placed in a Magnetic Field $\rightarrow$ <span style = "color:blue"> External Torque </span> $\rightarrow$ Zero of Potential Energy $\rightarrow$ Zero of Potential Energy </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Zero of Potential Energy </primary>
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- $\theta_1 = \frac{\pi}{2}$

- For an angle $\theta$ Oriantation

- **Potential energy**

- $U=-mB \cos \theta-\vec{m}.\vec{B}$

- Minimum PE $=-mB$ for $\theta =0$

- Maximum PE $=+mB$ for $\theta = \pi$

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<footer style = "font-size: 18px">Potential Energy of a Magnetic Dipole Placed in a Magnetic Field $\rightarrow$ External Torque $\rightarrow$ <span style = "color:blue"> Zero of Potential Energy </span> $\rightarrow$ Zero of Potential Energy $\rightarrow$ Zero of Potential Energy </footer>

### <Success style="font-size:25px"> Moving Coil Galvanometer Ammeter And Voltmeter L-7 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Radius R = 5cm

- Current I = 5A

- B = 0.1 T

- Magnetic moment

- Torque on the coil

- Change in PE when loop goes to minimum PE position

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<footer style = "font-size: 18px">Zero of Potential Energy $\rightarrow$ Zero of Potential Energy $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Thank You $\rightarrow$  </footer>