A particle mass m, charge Q, and kinetic energy T enter a transverse uniform magnetic field of induction $\vec

Question

A particle mass m, charge Q, and kinetic energy T enter a transverse uniform magnetic field of induction $\vec{B}$. After 3 sec the kinetic energy of the particle will be:

Accepted Answer

1. **Nature of Magnetic Force:** The magnetic force exerted on a moving charge is given by the Lorentz force formula: $\vec{F} = q(\vec{v} 	imes \vec{B})$. By definition of the cross product, this force is always perpendicular to the velocity vector ($\vec{v}$) of the particle .
2. **Work Done:** Since the force is perpendicular to the instantaneous displacement ($d\vec{l} = \vec{v}dt$), the work done by the magnetic force is zero:
   $$W = \int \vec{F} \cdot d\vec{l} = \int \vec{F} \cdot \vec{v} dt = 0$$ .
3. **Work-Energy Theorem:** According to the work-energy theorem, the change in kinetic energy is equal to the work done. Since $W = 0$, the change in kinetic energy $\Delta K = 0$.
4. **Conclusion:** The speed and kinetic energy of a charged particle moving in a magnetic field remain constant, even though its direction changes. Thus, the kinetic energy remains $T$ after any time interval .

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