Question

Compute the following Derivations:

1)Derive the cyclotron frequency of a point charge moving perpendicular to a constant magnetic field

2)Derive an expression for the charge on the capacitor as a function of time when a capacitor is being charged in an RC circuit

3)Derive an expression for the current as a function of time when a capacitor is being discharged through a resistor

Answer 1

Let us consider positive charged particle inside uniform magnetic field with velocity vector perpendicular to field . let B^rightarrow is inside the page and particle is moving in circular x path in plane perpendicular to B^rightarrow particle moves in circular path because force F_B is at 90 degree to velocity and B^rightarrow and has magnetic of qvB, f_B always points towards center of circle . therefore sigma f = ma f_e = qvB = mv^2/r therefore r = mv/qB therefore angular speed of particle is, w = v/r = v/mv qB w = qB/m where w rightarrow cyclotron frequency (2) consider the given circuit with resistor and capacitor in series and a battery as shown in figure capacitor is initially unchanged but starts to charge when the switch is closed potential difference across resistors is battery end and if steadily drops at capacitors charge, applying loop rule .