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). The 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

(a) Both magnetic force and electric force will act on the charge which will be perpendicular to each other for the orientation shown in figure.

Magnetic force will act towards right and electric field will act towards top.

Initially the body is still, so magnetic force is zero( as it depends on the magnitude of velocity)

So, it will move up with the direction of electric force initially.

(B)

As mentioned above

Initially the body is still, so magnetic force is zero.

So, it will move up with the direction of electric force initially.

After picking up speed in vertical direction, the magnetic force too increases and it act LS towards right.

With time the magnetic force will have velocity in both horizontal and vertically down direction.

So the charge initially moving towards top will curve towards right with time.

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