A charged particle moves through a magnetic field B with a velocity v. Which one of the following statements is true for the force (F) experienced by the particle?
F is maximum when v and B are perpendicular to each other.
Understanding Magnetic Force on a Charged Particle
When a charged particle moves through a magnetic field, it experiences a force known as the magnetic Lorentz force. The magnitude and direction of this force depend on the charge of the particle, its velocity, the magnetic field, and the angle between the velocity and the magnetic field.
The force (\(\mathbf{F}\)) experienced by a charged particle with charge \(q\), moving with velocity \(\mathbf{v}\) in a magnetic field \(\mathbf{B}\), is given by the formula:
\(\mathbf{F} = q(\mathbf{v} \times \mathbf{B})\)
This is a vector product. The magnitude of this force is given by:
\(F = |q| |\mathbf{v}| |\mathbf{B}| \sin\theta\)
or simply,
\(F = |q|vB \sin\theta\)
where:
- \(|q|\) is the magnitude of the charge of the particle.
- \(v\) is the magnitude of the velocity of the particle (speed).
- \(B\) is the magnitude of the magnetic field strength.
- \(\theta\) is the angle between the velocity vector (\(\mathbf{v}\)) and the magnetic field vector (\(\mathbf{B}\)).
The question asks when the force (F) experienced by the particle is maximum. From the formula \(F = |q|vB \sin\theta\), we can see that for given values of \(q\), \(v\), and \(B\), the magnitude of the force depends on the value of \(\sin\theta\).
To maximize \(F\), the value of \(\sin\theta\) must be maximum. The maximum possible value of \(\sin\theta\) is 1, which occurs when \(\theta = 90^\circ\) or \(\theta = 270^\circ\). A \(90^\circ\) angle means the velocity vector \(\mathbf{v}\) and the magnetic field vector \(\mathbf{B}\) are perpendicular to each other.
Let's evaluate the options based on this understanding:
- Option 1: F is maximum when v and B are parallel to each other.
If \(\mathbf{v}\) and \(\mathbf{B}\) are parallel, the angle between them is \(\theta = 0^\circ\). \(\sin(0^\circ) = 0\). The force magnitude is \(F = |q|vB \times 0 = 0\). This is the minimum force, not maximum.
- Option 2: F is maximum when v and B are anti-parallel to each other.
If \(\mathbf{v}\) and \(\mathbf{B}\) are anti-parallel, the angle between them is \(\theta = 180^\circ\). \(\sin(180^\circ) = 0\). The force magnitude is \(F = |q|vB \times 0 = 0\). This is also the minimum force, not maximum.
- Option 3: F is maximum when v and B are perpendicular to each other.
If \(\mathbf{v}\) and \(\mathbf{B}\) are perpendicular, the angle between them is \(\theta = 90^\circ\). \(\sin(90^\circ) = 1\). The force magnitude is \(F = |q|vB \times 1 = |q|vB\). Since the maximum value of \(\sin\theta\) is 1, this value of force is the maximum possible magnitude.
- Option 4: The force F is independent of the angle between v and B.
The formula \(F = |q|vB \sin\theta\) clearly shows that the force magnitude depends on \(\sin\theta\), and thus on the angle \(\theta\) between \(\mathbf{v}\) and \(\mathbf{B}\). This statement is incorrect.
Therefore, the force is maximum when the velocity of the charged particle is perpendicular to the magnetic field.
| Angle (\(\theta\)) between \(\mathbf{v}\) and \(\mathbf{B}\) | \(\sin\theta\) | Force (\(F = |q|vB \sin\theta\)) | Condition |
|---|---|---|---|
| \(0^\circ\) (Parallel) | \(0\) | \(0\) | Minimum Force |
| \(180^\circ\) (Anti-parallel) | \(0\) | \(0\) | Minimum Force |
| \(90^\circ\) (Perpendicular) | \(1\) | \(|q|vB\) | Maximum Force |
Revision Table: Magnetic Force Angle Dependence
This table summarizes how the magnetic force on a charged particle depends on the angle between its velocity and the magnetic field.
| Relative Direction of \(\mathbf{v}\) and \(\mathbf{B}\) | Angle (\(\theta\)) | \(\sin\theta\) | Magnitude of Force (\(F\)) | Result |
|---|---|---|---|---|
| Parallel | \(0^\circ\) | \(0\) | \(0\) | Minimum Force |
| Anti-parallel | \(180^\circ\) | \(0\) | \(0\) | Minimum Force |
| Perpendicular | \(90^\circ\) | \(1\) | \(|q|vB\) | Maximum Force |
| Any other angle | \(0^\circ < \theta < 180^\circ\) | \(0 < \sin\theta \le 1\) | \(0 < F \le |q|vB\) | Intermediate Force |
Additional Information: Lorentz Force and Direction
The magnetic force discussed here is part of the overall Lorentz force, which is the total force on a charged particle due to both electric and magnetic fields.
The Lorentz force \(\mathbf{F}_{\text{Lorentz}}\) is given by:
\(\mathbf{F}_{\text{Lorentz}} = q\mathbf{E} + q(\mathbf{v} \times \mathbf{B})\)
where \(\mathbf{E}\) is the electric field vector.
The direction of the magnetic force \(\mathbf{F} = q(\mathbf{v} \times \mathbf{B})\) is perpendicular to both the velocity \(\mathbf{v}\) and the magnetic field \(\mathbf{B}\). This direction can be determined using the right-hand rule for the cross product \(\mathbf{v} \times \mathbf{B}\). If the charge \(q\) is positive, the force is in the direction of \(\mathbf{v} \times \mathbf{B}\). If the charge \(q\) is negative, the force is in the opposite direction of \(\mathbf{v} \times \mathbf{B}\).
When the force is maximum (velocity is perpendicular to the magnetic field), the particle moves in a circular path if the magnetic field is uniform and perpendicular to the velocity.
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