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Physics / Chapter 25: Capacitance / Section 25-1

Capacitance — Question 6

Section 25-1 · Physics pages 4022–4023

Question 6 A prescribed charge-transfer rate gives a voltage ramp An ideal capacitor C receives signed charge at constant rate dqA/dt = I0 > 0 for 0 ≤t ≤T. Initially v(0) = −V0, with V0>0. t v Schematic; labels, not drawing lengths, specify the data. (a) Find qA(t) and v(t). (b) Find when the voltage changes sign and the condition for that event to occur during the transfer.
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Question 6 - Solution A prescribed charge-transfer rate gives a voltage ramp t v (a) Method and calculation qA(t) = qA(0) + I0t = −CV0 + I0t. v(t) = qA(t)/C = −V0 + I0t/C. (b) Interpretation and further calculation v(t∗) = 0 ⇒t∗= CV0/I0. t∗≤T ⇐⇒I0T ≥CV0. Physical check The charge-transfer rate is prescribed, so no circuit-resistance or transient model is needed.

Question 6

A prescribed charge-transfer rate gives a voltage ramp

An ideal capacitor C receives signed charge at constant rate dqA/dt=I0>0dq_A/dt=I_0>0 for 0≤t≤T0\le t\le T. Initially v(0)=−V0v(0)=-V_0, with V0>0.

Diagram shown in the original SVG worksheet.

  1. Find qA(t) and v(t).

  2. Find when the voltage changes sign and the condition for that event to occur during the transfer.

Question 6 - Solution

A prescribed charge-transfer rate gives a voltage ramp

Diagram shown in the original SVG worksheet.

(a) Method and calculation

qA(t)=qA(0)+I0t=−CV0+I0t.q_A(t)=q_A(0)+I_0t=-CV_0+I_0t.v(t)=qA(t)/C=−V0+I0t/C.v(t)=q_A(t)/C=-V_0+I_0t/C.

(b) Interpretation and further calculation

v(t*)=0⇒t*=CV0/I0.v(t_*)=0\Rightarrow t_*=CV_0/I_0.t*≤T⇔I0T≥CV0.t_*\le T\Longleftrightarrow I_0T\ge CV_0.

Physical check

The charge-transfer rate is prescribed, so no circuit-resistance or transient model is needed.

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