4.2 MAGNETIC FORCE

4.2.1 Sources and fields

Before we introduce the concept of a magnetic field \( \mathbf{B} \), we shall recapitulate what we have learnt in Chapter 1 about the electric field \( \mathbf{E} \). We have seen that the interaction between two charges can be considered in two stages. The charge \( Q \), the source of the field, produces an electric field \( \mathbf{E} \), where

\[ \mathbf{E}=Q \hat{\mathbf{r}} /\left(4 \pi \varepsilon_{0}\right) r^{2} \](4.1)

where \( \hat{\mathbf{r}} \) is unit vector along \( \mathbf{r} \), and the field \( \mathbf{E} \) is a vector field. A charge \( q \) interacts with this field and experiences a force \( \mathbf{F} \) given by

\[ \mathbf{F}=q \mathbf{E}=q Q \hat{\mathbf{r}} /\left(4 \pi \varepsilon_{0}\right) r^{2} \](4.2)

As pointed out in the Chapter 1, the field \( \mathbf{E} \) is not just an artefact but has a physical role. It can convey energy and momentum and is not established instantaneously but takes finite time to propagate. The concept of a field was specially stressed by Faraday and was incorporated by Maxwell in his unification of electricity and magnetism. In addition to depending on each point in space, it can also vary with time, i.e., be a function of time. In our discussions in this chapter, we will assume that the fields do not change with time.

Just as static charges produce an electric field, the currents or moving charges produce (in addition) a magnetic field, denoted by \( \mathbf{B (r)} \), again a vector field. It has several basic properties identical to the electric field. It is defined at each point in space (and can in addition depend on time). Experimentally, it is found to obey the principle of superposition: the magnetic field of several sources is the vector addition of magnetic field of each individual source.

4.2.2 Magnetic Field, Lorentz Force

Let us suppose that there is a point charge \( q \) (moving with a velocity \( \mathbf{v} \) and, located at \( \mathbf{r} \) at a given time \( t \)) in presence of both the electric field \( \mathbf{E (r)} \) and the magnetic field \( \mathbf{B (r)} \). The force on an electric charge \( q \) due to both of them can be written as

\[ \mathbf{F}=q[\mathbf{E}(\mathbf{r})+\mathbf{v} \times \mathbf{B}(\mathbf{r})] \equiv \mathbf{F}_{\text {electric }}+\mathbf{F}_{\text {magnetic }} \](4.3)

This force was given first by H.A. Lorentz based on the extensive experiments of Ampere and others. It is called the Lorentz force. You have already studied in detail the force due to the electric field. If we look at the interaction with the magnetic field, we find the following features.

1. It depends on \( q \), \( \mathbf{v} \) and \( \mathbf{B} \) (charge of the particle, the velocity and the magnetic field). Force on a negative charge is opposite to that on a positive charge.
2. The magnetic force \( q \mathbf{[ v × B ]} \) includes a vector product of velocity and magnetic field. The vector product makes the force due to magnetic field vanish (become zero) if velocity and magnetic field are parallel or anti-parallel. The force acts in a (sideways) direction perpendicular to both the velocity and the magnetic field. Its direction is given by the screw rule or right hand rule for vector (or cross) product as illustrated in Fig. 4.2.
3. The magnetic force is zero if charge is not moving (as then |\( \mathbf{v} \)|= 0). Only a moving charge feels the magnetic force.

4.2

The direction of the magnetic force acting on a charged particle.
(a) The force on a positively charged particle with velocity \( \mathbf{v} \) and making an angle \( q \) with the magnetic field \( \mathbf{B} \) is given by the right-hand rule.
(b) A moving charged particle \( q \) is deflected in an opposite sense to \( –q \) in the presence of magnetic field.

The expression for the magnetic force helps us to define the unit of the magnetic field, if one takes \( q \), \( \mathbf{F} \) and \( \mathbf{v} \), all to be unity in the force equation \( \mathbf{F} = q [ \mathbf{V×B} ] =q v B sin \theta \hat{\mathbf{n}} \) , where \( \theta \) is the angle between \( \mathbf{v} \) and \( \mathbf{B} \) [see Fig. 4.2 (a)]. The magnitude of magnetic field \( B \) is 1 SI unit, when the force acting on a unit charge (1 C), moving perpendicular to \( \mathbf{B} \) with a speed 1m/s, is one newton.

Dimensionally, we have \( [B] = [F/qv] \) and the unit of \( \mathbf{B} \) are Newton second / (coulomb metre). This unit is called tesla (\( T \)) named after Nikola Tesla (1856 – 1943). Tesla is a rather large unit. A smaller unit (non-SI) called gauss (=\( 10^{–4} \) tesla) is also often used. The earth’s magnetic field is about \( 3.6 × 10^{–5} \) T. Table 4.1 lists magnetic fields over a wide range in the universe.

Table 4.1 Order of Magnitudes of Magnetic Fields in a Variety of Physical Situations

4.2.3 Magnetic force on a current-carrying conductor

We can extend the analysis for force due to magnetic field on a single moving charge to a straight rod carrying current. Consider a rod of a uniform cross-sectional area \( A \) and length \( l \). We shall assume one kind of mobile carriers as in a conductor (here electrons). Let the number density of these mobile charge carriers in it be \( n \). Then the total number of mobile charge carriers in it is \( nlA \). For a steady current \( I \) in this conducting rod, we may assume that each mobile carrier has an average drift velocity \( \mathbf{v}_d \) (see Chapter 3). In the presence of an external magnetic field \( \mathbf{B} \), the force on these carriers is: \[ \mathbf{F}=(n l A) q \mathbf{v}_{d} \times \mathbf{B} \] where \( q \) is the value of the charge on a carrier. Now \( nq \mathbf{v}_d \) is the current density \( \mathbf{j} \) and \( |(nq \mathbf{v}_d)|A \) is the current \( I \) (see Chapter 3 for the discussion of current and current density). Thus,

\[ \begin{aligned} \mathbf{F} &=\left[\left(n q \mathbf{v}_{d}\right) l A\right] \times \mathbf{B}=[\mathbf{j} A l] \times \mathbf{B} \\ &=I \boldsymbol{l} \times \mathbf{B} \end{aligned} \](4.4)

where \( \boldsymbol{l} \) is a vector of magnitude \( l \), the length of the rod, and with a direction identical to the current \( I \). Note that the current \( I \) is not a vector. In the last step leading to Eq. (4.4), we have transferred the vector sign from \( \mathbf{j} \) to \( \boldsymbol{l} \).

Equation (4.4) holds for a straight rod. In this equation, \( \mathbf{B} \) is the external magnetic field. It is not the field produced by the current-carrying rod. If the wire has an arbitrary shape we can calculate the Lorentz force on it by considering it as a collection of linear strips \( d\boldsymbol{l}_j \) and summing \[ \mathbf{F}=\sum_{j} \mathrm{Id} \boldsymbol{l}_{j} \times \mathbf{B} \] This summation can be converted to an integral in most cases.

On Permittivity and Permeability

In the universal law of gravitation, we say that any two point masses exert a force on each other which is proportional to the product of the masses \( m_1 \), \( m_2 \) and inversely proportional to the square of the distance \( r \) between them. We write it as \( F = Gm_1m_2/r^2 \) where \( G \) is the universal constant of gravitation. Similarly, in Coulomb’s law of electrostatics we write the force between two point charges \( q_1 \), \( q_2 \), separated by a distance \( r \) as \( F = kq_1q_2/r^2 \) where \( k \) is a constant of proportionality. In SI units, \( k \) is taken as \( 1/4\pi\varepsilon \) where \( \varepsilon \) is the permittivity of the medium. Also in magnetism, we get another constant, which in SI units, is taken as \( μ/4\pi \) where \( \mu \) is the permeability of the medium.

Although \( G \), \( \varepsilon \) and \( \mu \) arise as proportionality constants, there is a difference between gravitational force and electromagnetic force. While the gravitational force does not depend on the intervening medium, the electromagnetic force depends on the medium between the two charges or magnets. Hence, while \( G \) is a universal constant, \( \varepsilon \) and \( μ \) depend on the medium. They have different values for different media. The product \( \varepsilon\mu \) turns out to be related to the speed v of electromagnetic radiation in the medium through \( \varepsilon\mu=1/v^2 \) .

Electric permittivity \( \varepsilon \) is a physical quantity that describes how an electric field affects and is affected by a medium. It is determined by the ability of a material to polarise in response to an applied field, and thereby to cancel, partially, the field inside the material. Similarly, magnetic permeability \( \mu \) is the ability of a substance to acquire magnetisation in magnetic fields. It is a measure of the extent to which magnetic field can penetrate matter.

Example 4.1

A straight wire of mass \( 200 g \) and length \( 1.5 m \) carries a current of \( 2A \). It is suspended in mid-air by a uniform horizontal magnetic field \( \mathbf{B} \) (Fig. 4.3). What is the magnitude of the magnetic field?

4.3

VIEW SOLUTION

From Eq. (4.4), we find that there is an upward force \( \mathbf{F} \), of magnitude \( IlB \),. For mid-air suspension, this must be balanced by the force due to gravity: \[ m g = I lB \] \[ \begin{aligned} B &=\frac{m g}{I l} \\ &=\frac{0.2 \times 9.8}{2 \times 1.5}=0.65 \mathrm{~T} \end{aligned} \]

Note that it would have been sufficient to specify \( m/l \), the mass per unit length of the wire. The earth’s magnetic field is approximately \( 4 × 10^{–5} T \) and we have ignored it.

Example 4.2

If the magnetic field is parallel to the positive y-axis and the charged particle is moving along the positive x-axis (Fig. 4.4), which way would the Lorentz force be for
(a) an electron (negative charge),
(b) a proton (positive charge).

4.4

VIEW SOLUTION

The velocity \( \mathbf{v} \) of particle is along the x-axis, while \( \mathbf{B} \), the magnetic field is along the y-axis, so \( \mathbf{v×B} \) is along the z-axis (screw rule or right-hand thumb rule).
So,
(a) for electron it will be along –z axis.
(b) for a positive charge (proton) the force is along +z axis.