Magnetic Field Due to a Current
Questions for Short Answer
1. An electric current flows in a wire from north to south. What will be the direction of the magnetic field due to this wire at a point east of the wire? West of the wire? Vertically above the wire? Vertically below the wire?
ANSWER: We stretch the thumb of the right hand along the current i.e. along the south. Now we curl our fingers to pass through the point where the direction of the magnetic field is required. The direction of fingers at that point gives the direction of the magnetic field.
According to this rule, we find that at a point east of the wire the direction of the magnetic field is upwards.
At a point west of the wire, the direction of the magnetic field is downward.
At a point vertically above the wire, the direction of the magnetic field is towards the west.
At a point vertically below the wire, the direction of the magnetic field is towards the east.
2. The magnetic field due to a long straight wire has been derived in terms of µₒ, i, and d. Express this in terms of εₒ, c, i, and d.
ANSWER: The magnetic field due to a long wire at a point is derived as,
B =µₒi/(2πd)
But µₒ = 1/εₒc²
Hence B =(1/2πεₒc²)i/d
3. You are facing a circular wire carrying an electric current. The current is clockwise as seen by you. Is the field at the center coming towards you or going away from you?
ANSWER: When facing a circular wire carrying an electric current, if the current is clockwise, the field at the center is going away from the viewer. It can be verified by the right-hand thumb rule.
4. In Ampere's law ∮B.dl =µₒi, the current outside the curve is not included on the right-hand side. Does it mean that the magnetic field B calculated by using Ampere's law gives the contribution of only the currents crossing the area bounded by the curve?
ANSWER: Though the currents outside the curve are not included on the right-hand side in Ampere's law ∮B.dl =µₒi, the magnetic field B calculated by using this law gives the contribution of all the currents crossing or not crossing the area bounded by the curve. In fact, the left-hand side values for all the currents not crossing the area bounded by the curve are zero.
5. The magnetic field inside a tightly wound, long solenoid is B = µₒni. It suggests that the field does not depend on the total length of the solenoid, and hence if we add more loops at the ends of a solenoid the field should not increase. Explain qualitatively why the extra added loops do not have a considerable effect on the field inside the solenoid.
ANSWER: Let us explain it by a diagram as below. 
Diagram for Q-5
Figure (a) shows a cross-section of a current-carrying loop. In the upper section of the loop, the current is going into the plane, hence the direction of the magnetic field will be clockwise around it. In the lower section, the current is coming out of it hence the magnetic field will be anticlockwise around it. At the midpoint between these two sections P, the direction of the magnetic field is the same and horizontal and hence add up.
Now consider a cross-section of a long solenoid in figure (b). Suppose in all the upper sections of the wire the current is going in, while in the lower sections current is coming out. Let us draw magnetic field lines around each wire that touches the axis of the solenoid. Consider the magnetic field at a point P on this axis. The maximum contribution of magnetic fields at this point is by only four sections of wire that are colored. Other loops of the solenoid at either end do not have any effect on the magnetic field at point P. So adding more loops at the ends does not increase the magnetic field inside the solenoid.
6. A long straight wire carries a current. Is Ampere's law valid for a loop that does not enclose the wire? That encloses the wire but is not circular?
ANSWER: Yes, Ampere's law is valid for a loop that does not enclose the wire. Since there is no current-carrying wire inside the loop, the right-hand side value will be zero. The left-hand side closed integral ∮B.dl will also be zero.
It is even valid when the loop enclosing the wire is not circular.
7. A straight wire carrying an electric current is placed along the axis of a uniformly charged ring. Will there be a magnetic force on the wire if the ring starts rotating about the wire? If yes, in which direction?
ANSWER: When the ring starts rotating, charges move in a circular motion and are equivalent to a loop, carrying current. Hence the magnetic field due to the ring at wire is along the wire. So the force on the wire, F =ilB.sinδ
Here δ =0 or π, So F =0.
So there will be no magnetic force on the wire.
8. Two wires carrying equal currents i each, are placed perpendicular to each other, just avoiding contact. If one wire is held fixed and the other is free to move under the magnetic forces, what kind of motion will result?
ANSWER: The magnetic field around the fixed wire will be anticlockwise if the current is coming towards the viewer. So for the other wire, the magnetic field on it will be perpendicular to the plane containing both wires but it will have opposite directions on either side of the fixed wire. See the diagram below.
Blue circle shows magnetic field around fixed wire
Red arrows are magnetic forces on the moving wire
Red arrows are magnetic forces on the moving wire
From the right-hand rule, the magnetic force on the moving wire will be in the plane containing both wires and also perpendicular to it. Its direction will be opposite on either side of the fixed wire. So the free wire will have a circular motion. Just the moment it rotates further after overlapping, the direction of current is opposite to the present in the free wire. Thus the direction of forces on it reverses and it rotates in the opposite direction. So the movement of the free wire will be periodic forward and reverse circular motion.
9. Two proton beams going in the same direction repel each other whereas two wires carrying currents in the same direction attract each other. Explain.
ANSWER: There is a marked difference between the two cases. The proton beams are net moving charges. Though these moving charges produce magnetic fields around it that have an attractive effect the electrostatic repulsion is predominant and the proton beams repel each other.
In a current-carrying wire, there is no net charge on the wire. Free or lose electrons move under an electric field created by a battery inside the wire. So there is an absence of electrostatic field outside the wire and only magnetic fields are present due to which two wires carrying current in the same direction attract each other.
10. In order to have a current in a long wire, it should be connected to a battery or some such device. Can we obtain the magnetic field due to a straight, long wire by using Ampere's law without mentioning this other part of the circuit?
ANSWER: Yes, we can obtain the magnetic field due to a long, straight wire by using Ampere's law without mentioning the battery or other devices. We only need to know the current in the wire. We take amperes loop enclosing this wire only.
11. Quite often, connecting wires carrying currents in opposite directions are twisted together in using electrical appliances. Explain how it avoids unwanted magnetic fields.
ANSWER: Wires carrying currents in the opposite directions have magnetic fields in counterclockwise directions. So if the wire on the left has magnetic fields in an anticlockwise direction and on the right wire has in a clockwise direction, after half a twist the wires exchange positions, and hence the direction of magnetic fields are also exchanged. So in a long stretch, the effect of the net magnetic field is zero.
12. Two current-carrying wires may attract each other. In absence of other forces, the wires will move towards each other increasing the kinetic energy. Does it contradict the fact that the magnetic force cannot do any work and hence cannot increase the kinetic energy?
ANSWER: When the magnetic force is acting on a moving charge in a magnetic field, no work is done by the magnetic force because it is always perpendicular to the displacement.
For other instances, the magnetic force can either do work if stored potential energy is converted or a supply of electrical energy is done through the appliances, for example, electric motor.
In the given case also, when the wires move towards each other, the moving electrons in the wire have a horizontal component of velocity which results in a magnetic force on them that opposes the drift velocity of electrons and reduces the current. To maintain the attraction between wires (and the current), the external sources have to increase the potential difference across each wire. So magnetic forces can not increase the kinetic energy itself, external energy has to be supplied for the purpose.
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