INVESTIGATION OF THE ENERGY STORED IN A CAPACITOR¶
Theory¶
The energy stored by a capacitor is given by the equation: $𝑈= 1/2QV$ Given that $𝑄=𝐶V$ then the equation for the energy stored can be written in the form: $𝑈=1/2 CV_{2}$. The capacitor can be charged to various values of $V$ and then the energy stored can be determined by using a Joule meter.
The energy stored can be measured as the capacitor discharges. A graph of energy stored against $V^{2}$ should be linear and the value of the capacitance can then be measured.
Apparatus:¶
- d.c. power supply
- Voltmeter (multimeter set on d.c. voltage range or CRO) – resolution ± 0.01V
- Digital joule meter
- 4mm leads
- Suitable switches
- Electrolytic capacitors e.g. a 1 000 μF or 2 200 μF
- Resistors e.g. 100 kΩ or other values
Experimental Method:¶
In [1]:
# Importing the necessary libraries
from matplotlib import pyplot as plt
import numpy as np
#from prettytable import PrettyTable
# Preparing the data to be computed and plotted
dt = np.array([
[1.0, 0.15],
[2.0, 0.29],
[3.0, 0.38],
[4.0, 0.50],
[5.0, 0.60],
[6.0, 0.81],
[7.0, 0.90],
[8.0, 1.13]
])
# Preparing X and y data from the given data
x = dt[:, 0].reshape(dt.shape[0], 1)
X = np.append(x, np.ones((dt.shape[0], 1)), axis=1)
y = dt[:, 1].reshape(dt.shape[0], 1)
# Calculating the parameters using the least square method
theta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(y)
print(f'The parameters of the line: {theta}')
# Now, calculating the y-axis values against x-values according to
# the parameters theta0 and theta1
y_line = X.dot(theta)
# Plotting the data points and the best fit line
plt.scatter(x, y)
plt.plot(x, y_line, 'r')
plt.title('Best fit line using regression method')
plt.xlabel('Length in cm')
plt.ylabel('Resistance')
plt.show()
#def makePrettyTable(table_col1, table_col2):
# table = PrettyTable()
# table.add_column("Column-1", x)
# table.add_column("Column-2", Y)
#return table
The parameters of the line: [[ 0.13452381] [-0.01035714]]
<Figure size 640x480 with 1 Axes>
Conclusion¶
In comparison....
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