The haplace's equation in spherical polar coordinates is :
$ [\frac{\partial}{\partial r} (\frac{1}{r^2} \frac{\partial U}{\partial r}) + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} (\sin \theta \frac{\partial U}{\partial \theta}) + \frac{1}{ \sin^2 \theta} \frac{\partial^2 U}{\partial \phi^2}] \frac{1}{r^2}= 0 $
The given question asks for the Laplace's equation in spherical polar coordinates. In mathematical physics, the Laplace equation is a second-order partial differential equation often encountered in various fields such as electrostatics, fluid dynamics, and quantum mechanics. Its general form is:
\(\nabla^2 U = 0\)where \(U\) is a scalar potential function.
In spherical polar coordinates (r, \theta, \phi), the Laplacian takes the form:
\[\nabla^2 U = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial U}{\partial r}\right) + \frac{1}{r^2 \sin \theta}\frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial U}{\partial \theta}\right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 U}{\partial \phi^2} = 0\]Analyzing the options given:
- Option 1: This considers the division by r^2, consistent with the Laplacian in this coordinate system.
- Option 2: Divides by r, which is not consistent with the spherical polar Laplacian.
- Option 3: Also divides by r, similarly incorrect.
- Option 4: Matches our derived expression for the Laplace's equation in spherical coordinates, with division by r^2.
Conclusion: The correct answer is Option 4, which accurately represents the Laplace equation in spherical polar coordinates:
\[\left[\frac{\partial}{\partial r} \left(\frac{1}{r^2} \frac{\partial U}{\partial r}\right) + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial U}{\partial \theta}\right) + \frac{1}{\sin^2 \theta} \frac{\partial^2 U}{\partial \phi^2}\right] \frac{1}{r^2}= 0\]Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A mass is attached to a spring that hangs vertically. The extension produced in the spring is 6 cm on Earth. The acceleration due to gravity on the surface of the Moon is one-sixth of its value on the surface of the Earth. The extension of the spring on the Moon would be:
Directions: Each item in this section consists of a sentence with an underlined word followed by four words (a), (b), (c), and (d). Select the option that is opposite in meaning to the underlined word and mark your response in your Answer Sheet accordingly.
The deluge affected the population.The major source of vitamins and minerals for vegetarians is
Which of the following statements about the Deccan Riots Commission is/are correct?
1. The Commission did not hold enquiries in the districts which were not affected.
2. The Commission did record the statements of ryots, sahukars and eye-witnesses.
Select the correct answer using the code given below:
