Define the terms: vector product, scalar field, vector field, gradient, divergence, curl, indicating which of the three latter apply to scalars and which to vectors. Prove that the necessary and sufficient conditions that a field of force \(\mathbf{F}\) be derivable (a) from a unique scalar potential (i.e. as its gradient), (b) from a unique vector potential (i.e. as its curl) are respectively: curl \(\mathbf{F}=0\), div \(\mathbf{F}=0\) throughout the space considered. Find expressions for the scalar and vector potentials of a magnetic particle of moment I.
A homogeneous sphere is set rotating about a horizontal axis. It is projected in the direction of this axis on a horizontal table. The coefficient of friction between the sphere and the table is \(\mu\). Discuss the subsequent motion.
Explain the method of inversion in electrostatic problems. Find an expression for the potential round an insulated charged conductor which consists of the larger parts of two equal spheres which cut one another in a re-entrant angle of 60 degrees.
A soap film is attached to fixed wires in the form of one or more closed curves. Assuming that the film takes such a form as to render its area a minimum consistent with the given boundary conditions prove, by considering the variation of the integral \[ \iint \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} dx dy, \] that the principal radii of curvature at any point of the film are equal in magnitude but opposite in sign. If the bounding wires are parallel circles whose planes are perpendicular to the line joining their centres prove that the meridian curve of the film is a catenary.
Find a differential equation which represents the path of a ray through a medium whose refractive index, \(\mu\), is a function of \(r\), the distance from a fixed centre. In the case when \(\mu = Cr^{-m}\), and \(C\) is a constant, show that the deviation of any ray in a portion of the path which subtends an angle \(\theta\) at the centre, is \(m\theta\).
An infinite circular cylinder of radius \(b\) and uniform density \(\sigma\) is surrounded by fluid of density \(\rho\). The outer boundary of the fluid is a concentric circular cylinder of radius \(a\). The outer cylinder is caused to execute small oscillations of amplitude \(\alpha\) in a direction perpendicular to its length. Show that the amplitude \(\beta\) of the oscillations of the inner cylinder is \[ \beta = \frac{2a^2\rho}{a^2(\sigma+\rho)+b^2(\sigma-\rho)}. \]