Moving Charges and Magnetism (12Acd05)

Sheet 02 Application of Biot Savart’s Law.

1. Calculate: the magnetic field induction at the center of a coil bent in the form of a square of side 2a carrying a current I.
   
   Ans: \(B=\frac{\sqrt{2}\mu_{o}I}{\pi a}\).
   
   
2. A long straight long conductor carrying current of \(30A\) is placed in an external uniform magnetic field of \(3\times10^{-4}T\) parallel to the current. Find: the magnitude and resultant magnetic field at a point \(3.0cm\) away from the wire.
   
   Ans: \(B=\left(3.6\times10^{-4}\right)T\).
   
   
3. Same current is flowing through three infinitely long wires along X,Y and Z directions. What is the magnetic field at a point \(\left(0,0,-a\right)\)?
   
   Ans: \(\vec{B}=\frac{\mu_{o}}{4\pi}\frac{I}{a}\left(\hat{j}-\hat{i}\right)\).
   
   
4. A circular coil of \(100\) turns of wire, of radius nearly \(20cm\) each, lies in X-Y plane with its center at the origin of the coordinates. Find: the magnetic field at the point \(\left(0,0,20\sqrt{3}cm\right)\) where this coil carries a current of \(\left(\frac{2}{\pi}\right)A\).
   
   Ans: \(B=25\mu T\).
   
   
5. Two semi-infinitely long straight current carrying conductors are held is the form as shown in below figure. One common end of them is at origin. If both the conductor carries the same current \(I\), find the value of magnetic field induction at point \(P\left(a,b\right)\).
   
   
   Ans: \(B=\frac{\mu_{o}I}{4\pi ab}\left[\left(a+b\right)+\sqrt{a^{2}+b^{2}}\right]T\).
   
   
6. A current of \(1.0A\) flowing in the sides of an equilateral triangle of side \(4.5\times10^{-2}m\). Find: the magnitude of the magnetic field at the centroid of the triangle.
   
   Ans: \(B=\left(4\times10^{-5}\right)T\).
   
   
7. Two parallel wires P and Q placed at a separation of \(r\left(r=5cm\right)\) carry electric current \(I_{1}=6A\) and \(I_{2}=3A\) in opposite direction as shown in the figure. Find: the point on the line PQ where the resultant magnetic field s Zero?
   
   
   Ans: \(x=5cm\).
   
   
8. Two current carrying conducting wires (1) and (2) are placed \(20cm\) apart on the same plane. Find: the magnitudes and directions of magnetic field at points P situated \(10cm\) away in the other side of wire (2) , Q in between wire (1) and (2), and R situated \(10cm\) away in the other side of wire (1)?Given: Wire (1) and (2) carries current in opposite direction.
   
   Ans: \(B_{R}=\left(4.67\times10^{-5}\right)T\).
   
   
9. The magnetic field due to a current carrying circular loop of radius \(12cm\) at its centre is \(1.5\times10^{-4}T\). Find: the magnetic field due to this loop at a point on the axis at a distance \(5cm\) from its center.
   
   Ans: \(\left(1.179\times10^{-4}\right)T\).
   
   
10. Calculate: the field at the centre of a semicircular wire of radius r as shown in below figure. if, straight wires are of infinite length.
    
    
    Ans: \(B=\frac{\mu_{o}I}{4\pi r}\left(\pi-2\right)\).
    
    
11. Two small identical circular coils marked as 1 and 2 carries an equal amount of current and are placed with their geometrical axis perpendicular to each other as shown in below figure. Derive an expression for the resultant magnetic field at O?
    
    
    Ans: \(B_{net}=\sqrt{2}\left(\frac{\mu_{o}}{4\pi}\right)\frac{2\pi I}{\left(a^{2}+x^{2}\right)^{\frac{3}{2}}}\).
    
    
12. Determine: the magnetic field at O due to current I through the circuit given below.
    
    
    Ans: \(B=\frac{\mu_{o}}{4\pi}I\theta\left[\frac{\left(R_{2}-R_{1}\right)}{R_{1}.R_{2}}\right]\).
    
    
13. The wire shown in the figure below carries a current of \(10A\) . Determine: the magnitude of the magnetic field at the center O. Given: radius of the bent portion is \(3cm\).
    
    
    
    Ans: \(B_{net}=\left(1.57\times10^{-4}\right)T\).
    
    
14. Two coaxial circular loops \(L_{1}\) and \(L_{2}\) of radii \(3cm\) and \(4cm\) are placed as shown in figure. What should be the magnitude and direction of the current though the loop \(L_{2}\) so that the net magnetic field at point O is Zero.
    
    
    Ans: \(I_{2}=0.56A\).
    
    
15. A current loop having two circular segments and joined by two radial lines. Find: the magnetic field at the centre O.
    
    
    Ans: \(B_{net}=\frac{\mu_{o}I\alpha}{4\pi}\left[\frac{\left(b-a\right)}{ab}\right]\).
    
    
16. The current loop PQRSTP formed by the two circular segments of radii \(R_{1}\) and \(R_{2}\) carries a current I Ampere. Find: the magnetic field at the common centre O.
    
    
    Ans: \(B=\frac{\mu_{o}I}{8}\left[\frac{1}{R_{2}}+\frac{3}{R_{1}}\right]\).
    
    
17. Two identical coils P and Q each of radius R are lying in perpendicular planes such that they have a common centre. Find: the magnitude and direction of the magnetic field at the common centre of the two coils, if they carry currents equal to \(I\) and \(\sqrt{3}I\)respectively.
    
    
    Ans: \(\frac{\mu_{o}I}{R}\) and \(\theta=30^{o}\).
    
    
18. In below figure the curved portion is semi-circular and the straight wire are long. Find: the magnitude of the magnetic field at the point O.
    
    
    Ans: \(B_{NET}=\frac{\mu_{o}I}{2d}\left(1+\frac{2}{\pi}\right)\).
    
    
19. A straight wire carrying a current of \(12A\) is bent in to a semi-circular arc of radius 2.0cm as shown in figure.(a) What is the direction and magnitude of \(\vec{B}\) at the center of the arc? (b) Would your answer change if the wire were bent in to a semi-circular arc of same radius but in opposite direction.
    
    
    Ans: \(B=\left(1.9\times10^{-4}\right)T\).
    
    
20. A long wire is bent as shown in below figure. What will be the magnitude and the direction of the magnetic field at the center O of the circular portion, if a current I is passed through the wire? Assume: that various portion of the wire do not touch at the point P.
    
    
    Ans: \(\frac{\mu_{o}}{2r}\left(1+\frac{1}{\pi}\right)\) and Along vertically upwards from plane of the paper.
    
    
21. A current \(I=5.0A\) flows along a thin wire shaped as as shown in below figure. The radius of curved part of the wire is equal to \(R=120mm\), the angle \(2\theta=90^{o}\). Find: the magnetic field induction at point O due to current flowing through the wire.
    
    
    Ans: \(\left(2.8\times10^{-5}\right)T\).
    
    
22. Figure shows the two semi-circular loops of radius \(R_{1}\) and \(R_{2}\) carrying a current I. Find: the magnitude and direction of the magnetic field at the common center O.
    
    
    Ans: \(\frac{\mu_{o}I}{4}\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\) and Normally downward from the plane of paper.
    
    
23. A current I ampere is flowing through the bent wire as shown in figure. Find: the magnitude and direction of the magnetic field at point O.
    
    
    Ans: \(B=\frac{\mu_{o}}{4\pi}\frac{I\alpha}{r}\) and Directed vertically downwards.
    
    
24. A current loop a b c d e formed by two circular segment of radii \(r_{1}=4cm\) and \(r_{2}=6cm\) with the common center at O. Carries a current of \(2A\). What is the magnitude of the field at the point O when \(\theta=90\)?
    
    
    Ans: \(B=\left(2.86\times10^{-5}\right)T\).
    
    
25. Two identical loops P and Q each of radius \(5cm\) are lying in perpendicular planes such that they have a common center as shown in figure. Find: the magnitude and direction of the net magnetic field at the common center of the two coils, if the coils are carrying currents equal to \(3A\) and \(4A\) respectively.
    
    
    Ans: \(B=\left(2\pi\mu\right)T\) and \(\theta=tan^{-1}\frac{4}{3}\).
    
    
26. Two identical circular loop of wires P and Q each of radius R carries a current I are kept in perpendicular planes such that they have common center as shown in below figure. Find: the magnitude and direction of the net magnetic field at the common center of the two wires?
    
    
    Ans: \(\frac{\mu_{o}I}{\sqrt{2}R}\) and at an angle of 45 degree with either of two fields.