Question

The force $\overrightarrow{\mathrm{F}}$ experienced by a particle of charge $\mathrm{q}$ moving with a velocity $\overrightarrow{\mathrm{v}}$ in a magnetic field $\overrightarrow{\mathrm{B}}$ is given by $\overrightarrow{\mathrm{F}}=\mathrm{q}(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})$. Which pairs of vectors are always at right angles to each other?

• A. $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{v}}$
• B. $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{B}}$
• C. $\overrightarrow{\mathrm{B}}$and $\overrightarrow{\mathrm{v}}$
• D. $\overrightarrow{\mathrm{F}}$and $(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})$

Answer 1

Answer: (A), (B)

$\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{B}}$

Explanation for the correct option:

Option (A): $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{v}}$

1. Here, The force $\overrightarrow{\mathrm{F}}$ experienced by a particle of charge $\mathrm{q}$ moving with a velocity $\overrightarrow{\mathrm{v}}$ in a magnetic field $\overrightarrow{\mathrm{B}}$ is given by $\overrightarrow{\mathrm{F}}=\mathrm{q}(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})$.
2. The pair $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{v}}$ always tend to be at a right angle to each other as $\mathrm{F}$is always perpendicular to the plane containing $\mathrm{B}\mathrm{and}\mathrm{v}$, however, Vectors $\mathrm{B}\mathrm{and}\mathrm{v}$ may have any angle between them.
3. This can be explained based on the right-hand rule, which determines the direction of the force and the magnetic field of a moving particle.
4. The right-hand rule by Fleming states that when the first finger (index finger), the second (middle) finger and the thumb of the right hand are placed perpendicularly (right angle) to each other, the thumb always indicates the direction of the velocity/direction of the moving particle, while the middle finger indicates the magnetic force, and the index finger always indicates the direction of the magnetic field.
5. Hence, using this rule, we can say that the force is said to be perpendicular to the velocity i.e $\mathrm{F}\perp \mathrm{v}$.

Option (B): $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{B}}$

1. Here, The force $\overrightarrow{\mathrm{F}}$ experienced by a particle of charge $\mathrm{q}$ moving with a velocity $\overrightarrow{\mathrm{v}}$ in a magnetic field $\overrightarrow{\mathrm{B}}$ is given by $\overrightarrow{\mathrm{F}}=\mathrm{q}(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})$.
2. The pair $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{B}$ always tend to be at a right angle to each other because $\mathrm{F}$ is always perpendicular to the plane containing $\mathrm{B}\mathrm{and}\mathrm{v}$, however, Vectors $\mathrm{B}\mathrm{and}\mathrm{v}$ may have any angle between them.
3. This can also be explained based on the right-hand rule, which states that when the first finger (index finger), the second (middle) finger and the thumb of the right hand should be placed perpendicularly (right angle) to each other to determine the direction of the force and the magnetic field of a moving particle.
4. Thus, by using the right-hand rule, the force is said to be perpendicular to the magnetic field i.e $\mathrm{F}\perp \mathrm{B}$.

Explanation for the incorrect option:

Option (C): $\overrightarrow{\mathrm{B}}$and $\overrightarrow{\mathrm{v}}$

1. The vectors $\overrightarrow{\mathrm{B}}$and $\overrightarrow{\mathrm{v}}$ are not perpendicular to each other.
2. They tend to have any angle between them, but they are not at the right angle to each other.
3. Hence, the given option is the incorrect one.

Option (D): $\overrightarrow{\mathrm{F}}$and $(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})$

1. $\overrightarrow{\mathrm{F}}$and $(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}})$ are not perpendicular to each other, hence are not at right angles to each other.
2. Hence, the given option does not satisfy the correct concept.

Thus, the pairs of vectors always at right angles to each other are: $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{v}}$,$\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{B}}$.

Hence options A and B are correct.