Physics · Ch 4 — Moving Charges and Magnetism

Magnetic Field, Lorentz Force

4.2.2

Magnetic Field, Lorentz Force

The Lorentz Force: Uniting Electric and Magnetic Effects

When a charged particle moves through a region containing both an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$ , the total force acting on it is the sum of the electric force and the magnetic force. This combined force is called the Lorentz force, named after H.A. Lorentz.

The total force $\mathbf{F}$ on a charge $q$ moving with velocity $\mathbf{v}$ at a point where the electric field is $\mathbf{E}$ and the magnetic field is $\mathbf{B}$ is given by:

\mathbf{F} = q \left[ \mathbf{E}(\mathbf{r}) + \mathbf{v} \times \mathbf{B}(\mathbf{r}) \right]

This can be split into two parts:

Electric force: $\mathbf{F}_\text{electric} = q \mathbf{E}$
Magnetic force: $\mathbf{F}_\text{magnetic} = q (\mathbf{v} \times \mathbf{B})$

Key Features of the Magnetic Force

The magnetic part, $q (\mathbf{v} \times \mathbf{B})$ , has several unique properties that distinguish it from the electric force:

Dependence on charge, velocity, and field: The magnetic force depends on the charge $q$ , the velocity $\mathbf{v}$ , and the magnetic field $\mathbf{B}$ . A negative charge experiences a force in the opposite direction to that on a positive charge.
Direction is perpendicular: Because it involves a cross product ( $\mathbf{v} \times \mathbf{B}$ ), the magnetic force always acts in a direction perpendicular to both the velocity $\mathbf{v}$ and the magnetic field $\mathbf{B}$ . Its direction is given by the right-hand rule (or screw rule) for cross products.
Zero force for parallel motion: If the velocity $\mathbf{v}$ is parallel or anti-parallel to the magnetic field $\mathbf{B}$ , the cross product $\mathbf{v} \times \mathbf{B}$ is zero. Therefore, the magnetic force is zero when a charge moves along the field lines.
Zero force on a stationary charge: If the charge is not moving ( $|\mathbf{v}| = 0$ ), the magnetic force is zero. Only a moving charge experiences a magnetic force.

Magnitude of the Magnetic Force

The magnitude of the magnetic force is given by:

|\mathbf{F}_\text{magnetic}| = q v B \sin \theta

where:

$q$ is the magnitude of the charge.
$v$ is the speed of the charge.
$B$ is the magnitude of the magnetic field.
$\theta$ is the angle between the velocity vector $\mathbf{v}$ and the magnetic field vector $\mathbf{B}$ .

The force is maximum when $\theta = 90^\circ$ (charge moves perpendicular to the field) and zero when $\theta = 0^\circ$ or $180^\circ$ (charge moves parallel or anti-parallel to the field).

Defining the Unit of Magnetic Field: The Tesla

The expression for the magnetic force is used to define the SI unit of magnetic field, the tesla (T). …

Figure 4.2The direction of the magnetic force acting on a charged particle. (a) The force on a positively charged particle with velocity v and making an angle θ with the magnetic field 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.

Fig. 4.2 — The direction of the magnetic force acting on a charged particle. (a) The force on a positively charged particle with velocity v and making an angle θ with the magnetic field 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.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your NCERT textbook's own diagram.

What Figure 4.2 Shows

The figure has two parts, (a) and (b), both illustrating the magnetic force on a moving charged particle.

Part (a) shows a perspective view of a horizontal plane. At the front-left of this plane is a positive charge $+q$ . A stylised right hand is drawn on the plane near the charge. Three vectors emerge from the charge:

Velocity $\mathbf{v}$ points to the lower-right, making an angle $\theta$ with the horizontal direction. The angle $\theta$ is marked between $\mathbf{v}$ and the magnetic field.
Magnetic field $\mathbf{B}$ points horizontally to the right.
Magnetic force $\mathbf{F}$ points straight upward (bold), perpendicular to both $\mathbf{v}$ and $\mathbf{B}$ .

The right hand is used to show the direction of $\mathbf{F}$ for a positive charge: if you point your fingers along $\mathbf{v}$ and curl them toward $\mathbf{B}$ , your thumb gives the direction of $\mathbf{F}$ .

Part (b) shows a vertical plane. Here, velocity $\mathbf{v}$ points straight upward. A positive charge $+$ experiences a magnetic force $\mathbf{F}$ in one direction (say, to the left), while a negative charge $-$ experiences the opposite force (to the right). The magnetic field $\mathbf{B}$ is horizontal (into or out of the page, depending on the drawing). This panel emphasises that the sign of the charge reverses the force direction.

Physical Idea Taught

The figure teaches the Lorentz magnetic force — the force on a moving charge in a magnetic field. Key features:

The force depends on the charge $q$ , velocity $\mathbf{v}$ , and magnetic field $\mathbf{B}$ .
It is given by the cross product $\mathbf{F} = q (\mathbf{v} \times \mathbf{B})$ , so the force is perpendicular to both $\mathbf{v}$ and $\mathbf{B}$ .
The magnitude is $F = |q| v B \sin\theta$ , where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{B}$ . If $\mathbf{v}$ is parallel or antiparallel to $\mathbf{B}$ ( $\theta = 0^\circ$ or $180^\circ$ ), the force is zero.
For a negative charge, the force direction is opposite to that for a positive charge (as shown in part (b)).
Only moving charges experience this force; stationary charges feel no magnetic force.

Key Formula

The textbook develops the Lorentz force expression:

\mathbf{F} = q [ \mathbf{E}(\mathbf{r}) + \mathbf{v} \times \mathbf{B}(\mathbf{r}) ]

For the magnetic part alone: …

Start your 14-day free trial to unlock the full solution →

← Previous · 4.2.1Sources and Fields Next · 4.2.3 →Magnetic Force on a Current-carrying Conductor