Please show detailed work. In the semi-classical model, an electron orbits a nuc

Question

Please show detailed work.

In the semi-classical model, an electron orbits a nucleus with an angular velocity omega_0, and the electric force is the only centripetal F_e = m_eW_^2_0r. When a magnetic field B = B_2 2 is applied normally to the orbital's plane, it is observed that the angular velocity changes to becomes omega, with no change in radius. Assume that the electron orbits counter-clockwise in relation to 2. Use Newton's second law in rtz to establish a simple relation between omega_3, omega_o, e, B and m_e, Write your answer in the form of a quadratic equation in omega but do not solve it. Substitute Omega_0 = omega - delta omega in the previous equation to show that delta omega only depends on e.B and me_e you may use the following approximation: (omega - delta omega)^2 or omega^2 - 2 omega delta omega

Explanation / Answer

Initial force will be

qE = mw02r ....................1

the additional force will take angular velocity to w

qE + Fb =  mw2r

From eq 1

Fb = mw2r +  mw02r

Fb = ewr B

ewr B = mw2r +  mw02r

ew B = mw2 +  mw02

mw2 + ew B - mw02 = 0

Using Wo = W + dW

mw2 + ew B - m(w + dw)2 = 0

mw2 + ew B - m (w2 - 2wdw)= 0

dw = eB/2m

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