question_answer

(i) Use Biot-Savart law to obtain a formula for magnetic field at the centre of a circular loop of radius R carrying a steady current I. Indicate the direction of the magnetic field.

(ii) A solenoid of length 5m has a radius of 1 cm and is made of 500 turns it carries a current of 5 A.

What is the magnitude of the magnetic field inside the solenoid.

Or

(i) A short bar magnet of magnetic moment \[=\text{ }32\text{ }A{{m}^{2}}\] is placed in a uniform magnetic field of 15 T. If the bar is free to rotate in the plane of field, which orientations would correspond to its

(a) stable and

(b) unstable equilibrium? What is the potential energy of the magnet in each case?

(ii) State Gauss' law in magnetism.

(iii) Define magnetic susceptibility of a material. Name two elements, one having positive susceptibility and other having negative susceptibility.

Answer:

(i) Consider a circular loop of radius R carrying current I. Magnetic field at its centre is given by According to Biot-Savart?s law, the magnetic field at the centre of the loop due to this current element,             \[dB=\frac{{{\mu }_{0}}}{4\pi }i\frac{dI\times r}{{{r}^{3}}}\] Since, the direction of \[dI\] is along the tangent, so\[d\,\,I\bot r.\] Thus,    \[|d\,\,B|=dB=\frac{{{\mu }_{0}}}{4\pi }\frac{idIr}{{{r}^{3}}}=\frac{{{\mu }_{0}}}{4\pi }\cdot \frac{idI}{{{r}^{2}}}\] \[\therefore \] Magnetic field at pointO due to complete loop,             \[\int{dB=B=\int_{0}^{2\pi r}{\frac{{{\mu }_{0}}}{4\pi }\frac{idI}{{{r}^{2}}}}}\]             \[=\frac{{{\mu }_{0}}}{4\pi }\cdot \frac{i2\pi r}{{{r}^{2}}}=\frac{{{\mu }_{0}}}{2}\frac{i}{r}\]             \[B=\frac{{{\mu }_{0}}}{2}\cdot \frac{i}{r}\] The direction of magnetic field at the centre of circular loop is given by right hand rule. As current carrying loop has the magnetic field lines around it which exerts a force on a moving charge. Thus, it behaves as a magnet with two mutually opposite poles. The anti-clockwise flow of current behaves like a North pole, whereas clockwise flow as South pole. Hence, loop behaves as a magnet. (ii) Given,\[l=5\,\,\text{m},\,\,r=1\,\,\text{cm}\]             \[N=500,\,\,I=5A\] \[\therefore \]Number of turns per unit length\[(n)=\frac{N}{l}\]             \[n=\frac{500}{5}=100\] Magnetic field inside the solenoid \[(B)={{\mu }_{0}}\,\,n\,\,I\] \[B=4\pi \times 10h-7\times 100\times 5=20\pi \times {{10}^{-5}}\,\,\text{T}\] Or (i) (a) For stable equilibrium Magnetic moment is parallel to B in this condition, the net force and torque on the bar magnet is zero,             \[F=0\,\,\text{and}\,\,\tau =0\] and its potential energy U is minimum \[\because \]       \[U=-M\cdot B=-MB\cos \theta \] \[\because \]       \[\theta =0\] \[\therefore \]      \[U=-MB\](minimum)\[=-32\times 15=-480\,\,\text{J}\] (b) For unstable equilibrium Magnetic moment is antiparallel to B, i.e. \[\theta =180{}^\circ \] \[U'=-MB\cos \theta =-32\times 15(-1)=480\,\,\text{J}\] (ii) Gauss?s law in Magnetism According to this law, the net magnetic flux \[({{\phi }_{B}})\] through any closed surface is always zero. i.e.        \[{{\phi }_{B}}=\oint{B\cdot dS=0}\] (iii) Magnetic susceptibility It is equal to the ratio of intensity of magnetisation (I) induced in the material to the magnetising force (H). It is denoted by \[{{\chi }_{m}}\]. i.e.        \[{{\chi }_{m}}=\frac{I}{H}\] Positive susceptibility \[\to \] paramagnetic substances, like aluminium, sodium. Negative susceptibility \[\to \] Diamagnetic substances, like copper, gold.