We understand the motion of a charged particle in a uniform electric field: usua

Question

We understand the motion of a charged particle in a uniform electric field: usually it is a straight line, but in general it is a parabola, just as masses follow parabolas in the presence of the Earth's uniform gravitational field. We also understand the motion of a charged particle in a uniform magnetic field: it is a circle, because the magnetic force is always perpendicular to the velocity vector and is thus the perfect centripetal force. What happens if a charged particle is placed at rest in the presence of both a uniform electric field and a uniform magnetic field, at right angles with each other? To be definite, suppose the electric field points up and the magnetic field points out of the page as shown below. Think carefully about how the electric and magnetic forces change as the proton picks up speed and begins to change direction. Figure out the shape of the path it will follow by thinking conceptually. Super bonus: Solve N2L exactly to find the path. Let E = (0, E, 0), B = (0,0, B), and write the velocity in components as v = (v_x, v_y, v_z). This initial condition is v_0 = 0: it: begins at rest. The total electric and magnetic force on the charge must equal its mass times its acceleration, so qE + qv times B = m dv/dt Solve each component of this equation separately. The problem is that the cross product will mix up x's and y's, so there will be some mathematical trickery required to figure this one out.

Explanation / Answer

If the particle is placed at rest in presence of uniform ekectric field then it will start accelerating it and particle will start moving in direction of field.

But in presence of magnetic field if particle is at rest then no force will act on it and particle will stay at its place.

When both fields are simultaneously applied then first due to electric field particle will start moving in field direction and with this it pick up some speed and then magnetic force will act on it due to which particle changes its direction.

Due to both forces acting the shape of the path will be circular.

The equation of motion have to be solved separately that is separate out x, y and z component and then solve by simple first order differential equation.

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