 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Natural sciences course of the University of Cambridge

Year of the natural sciences course of the University of Cambridge

Year 2 of the natural sciences course of the University of Cambridge

Physics course of the year 2 of the natural sciences course of the University of Cambridge

Physics A course of the University of Cambridge

Past exam of the physics A course of the University of Cambridge

2018

Paper 2

Section B

The capacitor-charge <complex amplitude> is
$$
q_0=\frac{V_0/L}{1/(LC)-\omega^2+i(R/L)\omega}.
$$
For fixed driving-voltage <amplitude>, maximizing $|q_0|$ minimizes
$$
D(\omega)=\left(\omega_0^2-\omega^2\right)^2+\gamma^2\omega^2,\qquad
\omega_0^2=\frac1{LC},\quad\gamma=\frac RL.
$$
Differentiate:
$$
\frac{dD}{d\omega}=2\omega[-2(\omega_0^2-\omega^2)+\gamma^2].
$$
The positive-frequency minimum therefore has
$$
\boxed{\omega_{\rm res}=\sqrt{\frac1{LC}-\frac{R^2}{2L^2}}},\qquad
R^2<\frac{2L}{C}.
$$
This is the <resonance> of charge, or capacitor voltage. If the stated inequality fails, their maximum is at zero <frequency>, and no positive displacement-resonance peak exists. The <electric current> has the additional factor $\omega$, and instead peaks exactly at $\omega_0=1/\sqrt{LC}$ for every $R>0$.

Solution

Past exam of the physics A course of the University of Cambridge / 2018 / Paper 2 / Section B / B6 / d / Solution

Past exam of the physics A course of the University of Cambridge / 2018 / Paper 2 / Section B / B6 / d / Solution

 Ancestors (15)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the physics A course of the University of Cambridge / 2018 / Paper 2 / Section B / B6 / d / Solution

 Ancestors (15)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)