Laplace Equation in Polar Coordinates
The given equation is Laplace's equation in Cartesian coordinates:
$ \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0 $
We need to transform this equation into polar coordinates ($r, \theta$), where $x = r \cos \theta$ and $y = r \sin \theta$.
Coordinate Transformation Formula
The Laplacian operator ($\nabla^2$) in polar coordinates is given by the formula:
$ \nabla^2 u = \frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} $
Transformed Equation
Setting the Laplacian in polar coordinates equal to zero gives the transformed Laplace equation:
$ \frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} = 0 $
This matches Option A.
Important Questions from Miscellaneous
- Which of the following relationships is/are not true?
(A). Most probable velocity = $\sqrt{\frac{2RT}{M}}$
(B). PV = $\frac{3}{2}kT$
(C). Compressibility factor Z = $\frac{pV}{nRT}$
(D). Average kinetic energy of gas = $\frac{1}{2}kT$
Choose the correct answer from the options given below - Match List-I with List-II
List-I List-II Electronic Configuration First Ionisation energy (kJ mol$^{-1}$) (A). ns$^2$ (I). 2100 (B). ns$^2$np$^1$ (II). 1400 (C). ns$^2$np$^3$ (III). 800 (D). ns$^2$np$^6$ (IV). 900
Choose the correct answer from the options given below: - The shielding constant of a 2p electron (calculated using Slater's rules) is
- Match List-I with List-II
List-I List-II Spectroscopy Property (A). Raman (I). Polarizability (B). FTIR (II). Dipole Moment (C). UV-Visible (III). Absorbance (D). NMR (IV). Spin
Choose the correct answer from the options given below: - The structure of protein comprises of:
(A). Primary structure of protein is associated with amino acids
(B). Secondary structure of protein is associated to peptides
(C). Tertiary structure of protein is associated with polypeptide chains
(D). Quaternary structure of protein is associated with polypeptide chains
Choose the correct answer from the options given below:
