Maharashtra Board Class 12 Physics Chapter 10 Magnetic Fields due to Electric Current PDF Download

10. Magnetic Fields Due To Electric Current

Can You Recall?

Do you know that a magnetic field is produced around a current carrying wire?

What is right hand rule?

Can you suggest an experiment to draw magnetic field lines of the magnetic field around the current carrying wire?

Do you know solenoid? Can you compare the magnetic field due to a current carrying solenoid with that due to a bar magnet?

Do You Know?

You must have noticed high tension power transmission lines, the power lines on the big tall steel towers. Strong magnetic fields are created by these lines. Care has to be taken to reduce the exposure levels to less than 0.5 milligauss (mG).

Try This

You can show that wires having currents passing through them, (a) in opposite directions repel and (b) in the same direction attract.

Hang two conducting wires from an insulating support. Connect them to a cell first as shown in Fig. 10.1 (a) and later as shown Fig. 10.1 (b), with the help of binding posts. You will notice that the wires in (a) repel each other and those in (b) come closer, i.e., they attract each other as soon as the current starts. The force in this experiment is certainly not of electrostatic origin, even through the current is due to the electrons flowing in the wires. The overall charge neutrality is maintained throughout the wire, hence the electrostatic forces are ruled out.

10.1 Introduction

In this Chapter you will be studying how magnetic fields are produced by an electric current. Important foundation for further developments will also be laid down.

Hans Christian Oersted first discovered that magnetic field is produced by an electric current passing through a wire. Later, Gauss, Henry, Faraday and others showed that magnetic field is an important partner of electric field. Maxwell's theoretical work highlighted the close relationship of electric and magnetic fields. This resulted into several practical applications in day today life, for example electrical motors, generators, communication systems and computers.

In electrostatics, we have considered static charges and the force exerted by them on other charge or test charge. We now consider forces between charges in motion.

You have learnt in Xth Std. that if a magnetic needle is held in close proximity of a current carrying wire, it shows the direction of magnetic field circling around the wire. Imagine that a current carrying wire is grabbed with your right hand with the thumb pointing in the direction of the current, then your fingers curl around in the direction of the magnetic field (Fig. 10.2).

Teacher's Note

In India, power transmission lines carry high voltage electricity across long distances. The magnetic field around these wires can affect nearby areas, which is why safety distances are important.

Exam Trick

Remember the Right Hand Thumb Rule: Thumb shows current direction, fingers curl in magnetic field direction. Like screwing a bolt clockwise!

Points to Remember

Magnetic fields are produced by moving electric charges (currents).
The Right Hand Thumb Rule tells you the direction of the magnetic field around a current-carrying wire.
Parallel wires with currents in the same direction attract each other.
Parallel wires with currents in opposite directions repel each other.
The magnetic field gets weaker as you move away from the wire.

10.2 Magnetic Force

From the above discussion and Fig. 10.3, you must have realized that the directions of \(\vec{v}\), \(\vec{B}\) and \(\vec{F}\) follow a vector cross product relationship. Actually the magnetic force \(F_m\) on an electron with a charge -e, moving with velocity \(\vec{v}\) in a magnetic field \(\vec{B}\) is

\(\vec{F}_m = -e(\vec{v} \times \vec{B})\) --- (10.1)

In general for a charge q, the magnetic force will be

\(\vec{F}_m = q(\vec{v} \times \vec{B})\) --- (10.2)

If both electric field \(\vec{E}\) and the magnetic field \(\vec{B}\) are present, the net force on charge q moving with the velocity \(\vec{v}\) in

\(\vec{F} = q[\vec{E} + (\vec{v} \times \vec{B})]\) --- (10.3)

\(= q\vec{E} + q(\vec{v} \times \vec{B}) = \vec{F}_e + \vec{F}_m\) --- (10.4)

Justification for this law can be found in experiments such as the one described in Fig. 10.1 (a) and (b). The force described in Fig. (10.4) is known as Lorentz force. Here \(\vec{F}_e\) is the force due to electric field and \(\vec{F}_m\) is the force due to magnetic field.

There are interesting consequences of the Lorentz force law.

(i) If the velocity \(\vec{v}\) of a charged particle is parallel to the magnetic field \(\vec{B}\), the magnetic force is zero.

Thus the vectors \(\vec{v}\) and \(\vec{F}\) are always perpendicular to each other. Hence. \(\vec{F} \cdot \vec{v} = 0\), for any magnetic field \(\vec{B}\). Magnetic force \(\vec{F}_m\) is thus perpendicular to the displacement and hence the magnetic force never does any work on moving charges.

The magnetic forces may change the direction of motion of a charged particle but they can never affect the speed.

Interestingly, Eq. (10.2) leads to the definition of units of \(\vec{B}\). From Eq. (10.2),

\(\vec{F} = q|\vec{v} \times \vec{B}| = qvB \sin \theta\) --- (10.5)

where \(\theta\) is the angle between \(\vec{v}\) and \(\vec{B}\) and is unit vector in the direction of force.

If the force F is 1 N acting on the charge of 1 C moving with a speed of 1m s\(^{-1}\) perpendicular to \(\vec{B}\), then we can define the unit of B.

\(\therefore B = \frac{F}{qv}\)

\(\therefore\) unit of B is \(\frac{N \cdot s}{C \cdot m}\)

Dimensionally,
\([B] = [F/qv]\)

(ii) If the charge is stationary, \(\vec{v} = 0\), the force = 0, even if \(\vec{B} \neq 0\).

From Eq. (10.4) it may be observed that the force on the charge due to electric field depends on the strength of the electric field and the magnitude of the charge. However, the magnetic force depends on the velocity of the charge and the cross product of the velocity vector \(\vec{v}\) the magnetic field vector \(\vec{B}\), and the charge q.

Consider the vectors \(\vec{v}\) and \(\vec{B}\) with certain angle between them. Then \(\vec{v} \times \vec{B}\) will be a vector perpendicular to the plane containing the vectors \(\vec{v}\) and \(\vec{B}\) (Fig. 10.4).

Teacher's Note

The Lorentz force is why electric motors work. When current flows through a wire in a magnetic field, the wire moves. This is how fans and motors in your home work!

Exam Trick

Remember: Magnetic force is always perpendicular to velocity. So it cannot change the speed of a particle, only its direction. Like pushing a ball sideways while it rolls!

Points to Remember

Lorentz force is the total force on a moving charged particle in electric and magnetic fields.
Magnetic force depends on velocity, charge, and magnetic field strength.
Magnetic force is perpendicular to both velocity and magnetic field.
Magnetic force never does work on a charged particle.
The SI unit of magnetic field is tesla (T).

10.3 Cyclotron Motion

In a magnetic field, a charged particle typically undergoes circular motion. Figure 10.5 shows a uniform magnetic field directed perpendicularly into the plane of the paper (parallel to the -ve z axis).

Figure 10.5 shows a particle with charge q moving with a speed v, and a uniform magnetic field \(\vec{B}\) is directed into the plane of the paper. According to the Lorentz force law, the magnetic force on the particle will act towards the centre of a circle of radius R, and this force will provide centripetal force to sustain a uniform circular motion.

Thus

\(qvB = \frac{mv^2}{R}\) --- (10.6)

\(\therefore mv = p = qBR\) --- (10.7)

Equation (10.7) represents what is known as cyclotron formula. It describes the circular motion of a charged particle in a particle accelerator, the cyclotron.

Let us look at a charged particle which is moving in a circle with a constant speed. This is uniform circular motion that you have studied earlier. Thus, there must be a net force acting on the particle, directed towards the centre of the circle. As the speed is constant, the force also must be constant, always perpendicular to the velocity of the particle at any given instant of time. Such a force is provided by the uniform magnetic field \(\vec{B}\) perpendicular to the plane of the circle along which the charged particle moves.

Teacher's Note

In hospitals, cyclotron machines help treat cancer patients. The machine accelerates particles to high speeds which then target cancer cells. This is real science helping real people in India!

Exam Trick

Remember the cyclotron formula: \(mv = qBR\). The radius of the circular path increases with momentum but decreases with stronger magnetic field, just like a stronger curve needs more turning!

Points to Remember

A charged particle moves in a circular path in a uniform magnetic field perpendicular to its velocity.
The centripetal force comes from the magnetic force.
The cyclotron formula relates momentum to the radius of circular motion.
Cyclotron frequency is independent of particle energy and velocity.
Cyclotrons are used to accelerate particles for medical and research purposes.

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