Moving Charges and Magnetism (12Acd05)

Sheet 08 Higher Order Thinking Skills.

1. A long wire carrying current is bent into a circular coil of one turn and then into a circular coil of smaller radius having n turns. If, B is the magnetic field at the center of coil and same current is passed in both cases. Then, what will be the magnetic field at the center of the coil of n turns.
   
   Ans: \(B^{‘}=n^{2}\times B\).
   
   
2. Show that; the magnetic field along the axis of a current carrying circular circular coil of radius R at a distance X from the center O of this coil has a fraction decrease in the magnetic field is \(1.5=\frac{X^{2}}{R^{2}}\) .
   
   Ans: \(\frac{B’}{B}=1.5\times\frac{x^{2}}{R^{2}}\).
   
   
3. A pair of stationary and infinitely long bent wires are placed in the X-Y plane as shown in figure. The two wires carry current of 4A and 9A respectively. If, the segment D and A along X-axis and segment C and F are parallel to the Y-axis such that OB=2cm and OE=3cm, then calculate: the magnitude of magnetic induction at the origin O.
   
   
   Ans: \(B_{Net}=\left(5\times10^{-5}\right)T\).
   
   
4. An arrangement of three parallel straight wires placed perpendicular to the plane of the paper carrying same current I along the same direction as in given figure. Calculate: magnitude of the force per unit length on the middle wire B.
   
   
   Ans: \(\frac{F_{net}}{L}=\frac{\mu_{o}I^{2}}{4\pi a}\).
   
   
5. A current I is flowing through an infinitely long conductor bent into the as shown in the below diagram. If, the radius of the curved path is R. Find: magnetic field at the center O.
   
   
   Ans: \(\frac{\mu_{o}I}{4\pi R}\left(\frac{3\pi}{2}+1\right)\) acting vertically outward.
   
   
6. Using Biot Savart’s Law obtain an expression for the magnetic field at the center of a coil bent into a form of a square of side 2a carrying current I.
   
   Ans: \(B=\frac{\sqrt{2}\mu_{o}I}{\pi a}\).
   
   
7. A current of \(1.0A\) is flowing in the sides of an equilateral triangle of side \(4.5\times10^{-2}\). Find: the magnetic field at the centroid of the triangles.
   
   Ans: \(B=\left(4\times10^{-5}\right)T\).
   
   
8. Two semi-infinitely long straight current carrying conductors are held at right angle to each other so that their common end lies at origin, as shown in below figure, If, both the conductors carry the same current I as shown, find the magnetic field at the point \(P\left(a,b\right)\) .
   
   Ans: \(B=\frac{\mu_{o}I}{4\pi a b}\left[\sqrt{a^{2}+b^{2}}+\left(a+b\right)\right]\).
   
   
9. A galvanometer of resistance G is converted into a voltmeter to measure up to V volts by connecting a resistance \(R_{1}\) in series with the coil. If, a resistance \(R_{2}\) is connected in series with it, then it can measure upto \(\frac{V}{2}\) volts. Find: the resistance in terms of \(R_{1}\) and \(R_{2}\) required to be connected to convert into a voltmeter can read up to \(\) 2V. Also; find the resistance G of the galvanometer in terms of \(R_{1}\) and \(R_{2}\).
   
   Ans: \(R_{3}=3R_{1}-2R_{2}\) and \(G=R_{1}-2R_{2}\).
   
   
10. A voltmeter of resistance \(R_{v}\) and an ammeter of resistance \(R_{a}\) are connected in a circuit to measure a resistance R as shown in below figure. The ration of meter readings gives an apparent resistance \(R’\). Show that: \(R\) and \(R’\) are related by relation \(\frac{1}{R}=\frac{1}{R’}-\frac{1}{R_{v}}\) .
    
    
    Ans: \(\frac{1}{R}=\frac{1}{R’}+\frac{1}{R_{V}}\).
    
    
11. Find the magnitude of the force on each segment of the wire as shown in below figure. If, a magnetic field of \(0.03T\) , is applied parallel to the AB and DE. Take the value of current flowing through the wire is I ampere.
    
    
    Ans: \(F_{AB}=0\), \(F_{BC}=0.024N\), \(F_{CD}=0.015N\).
    
    
12. A close loop PQRS carrying a current is placed in a uniform magnetic field. If, the magnetic force on segment PS, SR and RQ are \(F_{1}\) , \(F_{2}\) and \(F_{3}\) respectively and are in the plane of the paper and along the direction shown in figure below. Then calculate the force acting on the segment QP.
    
    
    Ans: \(F_{PQ}=\sqrt{\left(F_{3}-F_{1}\right)^{2}+F_{2}^{2}}\).
    
    
13. The electric current in a circular coil of two turns produce a magnetic field of \(0.2T\) at its center. The coil is now unwound and is rewound in to a circular coil of 4 turns. Calculate: the magnetic field at the center of coil now is in tesla if same current flows in the coil.
    
    Ans: \(B_{2}=0.8T\).
    
    
14. A moving coil galvanometer has \(150\) equal divisions. Its current sensitivity is \(10div.mA^{-1}\) and voltage sensitivity is \(2div.mV^{-1}\). In order that each division reads \(1V\) . Calculate: the value of resistance (in ohm) needed to be connected in series with the coil.
    
    Ans: \(R-9.995\Omega\).
    
    
15. An electron of charge -e, mass m, enters a region of uniform magnetic field, \(\vec{B}=B\hat{i}\) with a velocity \(\vec{v}=v_{x}\hat{i}+v_{y}\hat{j}\) . What is the velocity of the electron after a time interval of t seconds?
    
    Ans: \(\vec{V}=v_{x}\hat{i}+v_{y}\hat{j}+\left(\frac{e}{m}v_{y}Bt\right)\hat{k}\).
    
    
16. A loop carrying a current \(I\) lies in the x-y plane as shown in figure. The unit vector along the direction coming perpendicularly from the plane of the screen is \(\hat{k}\), Calculate: the magnetic moment of current loop.
    
    
    Ans: \(\vec{M}=Ia^{2}\left[1+\frac{\pi}{2}\right]\hat{k}\).
    
    
17. The current flowing in galvanometer (G) when key \(K_{2}\) is kept open is \(I\). On closing the key \(K_{2}\), current through the galvanometer becomes \(\frac{1}{n}\), where n is an integer. Obtain an expression for resistance G of the galvanometer in terms of R, S, and n. To what form does this expression reduce when the value of R is very large as compared to S?
    
    
    Ans: \(G=\frac{\left(n-1\right)RS}{\left[R-\left(n-1\right)S\right]}\) and \(G=\left(n-1\right)S\).