Work done in a Capacitor

Capacitor is an electrical component that stores charge. As charges are stored, potential energy in the capacitor also increases. In this post, I would like to explain how to calculate the energy stored in a capacitor.

Question

The switch is closed at time t = 0 s. During the time interval from t = 0.0 s to t = 4.0 s, 15 mC charge passes through the resistor.

Calculate the energy transferred by the battery during this 4.0 seconds and the energy stored in the capacitor after 4.0 s.

Capacitor Circuit

Definition of Potential Difference

In many textbooks, we learnt that the potential difference across a component is the energy released for every unit charge passing through it.

V=\frac{W}{Q}

If we are to calculate the energy released by the battery, it would be

W=V \times Q = 9.0 \times 15 \times 10^{-3} =0.135 \text{J}

Energy stored in the capacitor

Image 2

However, the energy stored in the capacitor cannot be 0.135 J. This is because a capacitor’s potential difference changes as it stores charge. The equation above requires that the p.d. remains constant. Hence, we need to use another formula.

Since capacitance is defined as the charge stored for every unit potential difference,

Q=VC

i.e. the charge stored is directly proportional to the potential difference across it. Since the work done is the area under a charge-voltage graph, we will use the formula for a triangle.

W=\frac{1}{2}QV

At time t = 4.0 s,

V=\frac{15 \times 10^{-3}}{2000 \times 10^{-6}}=7.5 \text{V}

Hence,

W=\frac{1}{2} \times 15 \times 10^{-3} \times 7.5 = 0.05625 \text{J}

Lost energy?

The energy stored in the capacitor(0.05625 J) is not the same as the energy transferred from the battery(0.135 J) because heat is lost through the resistor as current flows.

 

Radioactive Decay is Random

Radioactive decay is both random over space and time, and radioactive decay is spontaneous.

Random

Random over space means that given a large number of radioactive nuclei, it would be impossible to predict which nuclei would decay next. Random over time means that you cannot predict when a nuclei would decay at any time.

Given a large number of nuclei, it can be safe to predict that approximately half of the original radioactive nuclei would decay after one half-life. However, the exact number of nuclei decay would not be identical given two samples of equal number of original radioactive nuclei.

Continue reading “Radioactive Decay is Random”

Ultrasound Calculations

The table provides the data for the acoustic impedance and absorption coefficients for muscle and bone.

ultrasound 4

A parallel beam of ultrasound of intensity I_0  enters the muscle of thickness 3.0 cm as shown. The ultrasound is then reflected at the tissue boundary and returns to the surface of the muscle.

Calculate the intensity, in terms of I_0 , that is received when the ultrasound returns back to the surface of the muscle.


There are three important parts to solve this problem. The first part would be the attenuation of intensity inside the muscle. The second part would be the reflection of ultrasound at the muscle-bone boundary. The last part would be the intensity attenuation inside the muscle back to the surface.

I=I_0 e^{-21 times 0.03}