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1: charge is stationary i.e. v=0
2: when charge is moving in the direction of magnetic field or in opposite direction i.e. θ=0 or 180 and sin θ =0.
Further, if only electrical field exists, then only electrical force applies on the point charge and the point charge is accelerated in the direction of electrical field ( E ). If only magnetic field exists, then only magnetic force applies on the point charge except for the cases mentioned above (when magnetic force is zero) and the point charge is accelerated in the direction of vector expression q( vXB ). If both electrical and magnetic field exist, then charge is subjected to both kinds of force provided conditions for zero magnetic force are not met. In the last case, acceleration of the point charge is in the direction of resultant force :
$$\mathbf{a}=\frac{\mathbf{F}}{m}=\frac{q[\mathbf{E}+\left(\mathbf{v}X\mathbf{B}\right)]}{m}$$
Problem : An electron, moving along x-axis in an uniform magnetic field B , experiences maximum magnetic force along z-axis. Find the direction of magnetic field.
Solution : Since the particle experiences maximum magnetic force, the angle between velocity and magnetic field vector is right angle. Now, magnetic force in z-direction is also perpendicular to the magnetic field. Hence, magnetic field is either in positive or negative y-direction. By applying Right hand rule of vector cross product, we find that it is oriented in positive y-direction if the charge is positive. But, charge on electron is negative.
Hence, magnetic field is oriented along negative y-direction.
The nature of magnetic force is different to electrical force. First, it is not linear in the sense that it does not operate in the direction of magnetic field. This is unlike electric force which acts in the direction of applied electric field. The magnetic force, as we have seen in the preceding section, acts in the side-way direction following vector cross product rule. Also, magnetic force is relatively weaker as magnetic field is a weaker field in comparison with electric field.
The first of the two distinguishing characteristics as described above has important implications. Since magnetic force is perpendicular to the direction of velocity, it can only change the direction of motion – not its magnitude. The magnetic force can not change the magnitude of velocity i.e. speed of the charged particle. In turn, we can say that magnetic force can not bring about a change in the kinetic energy of the charged particle as speed remains same due to magnetic field.
An immediate fall out of the magnetic force is very interesting. This force does no work. We know work is scalar dot product of force and displacement. Now, velocity is time rate of displacement. It means velocity and displacement have same direction. Since magnetic force is perpendicular to velocity, it is also perpendicular to small elemental displacement. What it means that magnetic force is always perpendicular to displacement. Thus, work done by magnetic force is zero.
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