To determine the correct equation for calculating the magnetic force on a charge moving in a magnetic field, let's analyze the options provided.
The magnetic force on a charge [tex]\( q \)[/tex] moving with velocity [tex]\( v \)[/tex] in a magnetic field [tex]\( B \)[/tex] is given by the Lorentz force law, which states:
[tex]\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \][/tex]
Here, [tex]\( \mathbf{v} \times \mathbf{B} \)[/tex] is the cross product of the velocity vector [tex]\( \mathbf{v} \)[/tex] and the magnetic field vector [tex]\( \mathbf{B} \)[/tex]. The magnitude of the force can be expressed as:
[tex]\[ F = |q| v B \sin \theta \][/tex]
Where:
- [tex]\( |q| \)[/tex] is the magnitude of the charge
- [tex]\( v \)[/tex] is the speed of the charge
- [tex]\( B \)[/tex] is the magnitude of the magnetic field
- [tex]\( \theta \)[/tex] is the angle between the velocity vector and the magnetic field vector
This equation essentially tells us that the magnetic force depends on the perpendicular component of the velocity relative to the magnetic field.
Now, let's evaluate the given choices based on this understanding:
1. [tex]\( F=|q| v B \cos \theta \)[/tex] - This equation uses cosine instead of sine, which is incorrect for calculating the magnetic force.
2. [tex]\( F=|q| v B \sin \theta \)[/tex] - This equation correctly represents the magnetic force on a charge moving in a magnetic field.
3. [tex]\( F=|q| B \cos \theta \)[/tex] - This equation is missing the velocity component [tex]\( v \)[/tex] and uses cosine instead of sine, so it is incorrect.
4. [tex]\( F=|q| B \sin \theta \)[/tex] - This equation is missing the velocity component [tex]\( v \)[/tex], so it is also incorrect.
Therefore, the correct equation to calculate the magnetic force on a charge moving in a magnetic field is:
[tex]\[ F=|q| v B \sin \theta \][/tex]
Hence, the correct choice is the second one.
