Part a
Calculate the magnitude of the induced electromotive force (emf), \(\varepsilon\), in the loop as it enters the magnetic field.
Part b
Using the induced emf from part (a) and the given resistance \(R = 4.0\,\Omega\), determine the magnitude of the induced current \(I\) in the loop.
Part b
Using your expression for \(v^2\) from part (a), show that the total mechanical energy \(E_{tot}\) of the satellite is given by \(E_{tot} = -\frac{1}{2}\frac{GMm}{r}\).
Part a
The gravitational force provides the centripetal force for the satellite’s circular orbit. By equating the expressions for gravitational force \(F_{g} = \frac{GmM}{r^2}\) and centripetal force \(F_{c} = \frac{mv^2}{r}\), show that the orbital speed \(v\) of the satellite is given by \(v = \sqrt{\frac{GM}{r}}\).
Part b
The orbital period \(T\) is the time taken for one complete orbit and is related to the orbital speed \(v\) and radius \(r\) by \(T = \frac{2\pi r}{v}\). Using your result from part (a), show that the orbital period is given by \(T^2 = \frac{4\pi^2}{GM}r^3\).
Part a
Derive an expression for the constant vertical acceleration \(a_y\) of a particle with charge \(q\) and mass \(m\) in a uniform electric field of magnitude \(E\) directed vertically downward. Assume the only vertical force is the electric force and use a coordinate system where y is positive downwards.
Part b
Using the result from part (a), determine the vertical deflection \(d\) (the y-displacement) and the final vertical velocity \(v_y\) of the particle after it travels a horizontal distance \(L\). Assume the initial vertical velocity is zero and the initial horizontal velocity is \(v_0\).