Find the electrical image of an external point charge in an uninsulated conducting sphere. Two conducting spheres have radii \(a\) and \(b\) each of which is small in comparison with \(c\) the distance between their centres. Show that the coefficients of potential \(p_{11}, p_{12}, p_{22}\) are given by the approximate equations \[ p_{11} = \frac{1}{a}, \quad p_{12} = \frac{1}{c}, \quad p_{22}=\frac{1}{b}, \] wherein the fourth and higher powers of the ratio of the larger radius to \(c\) are neglected.
Prove that the mutual potential energy of two small magnets of moments \(\mu, \mu'\), whose centres are at a distance \(r\) apart, is \[ \frac{\mu\mu'}{r^3}(\cos\epsilon - 3\cos\theta\cos\theta'), \] where \(\epsilon\) is the angle between the axes, and \(\theta, \theta'\), the angles the axes make with the line of centres. Two small magnets of equal moment \(\mu\) can turn freely about their centres which are fixed at a distance \(r\) apart. If there is a magnetic field of uniform intensity \(H\) perpendicular to the line of centres, show that the magnets can rest in stable equilibrium with their axes in the direction of the field, provided \[ H > \frac{3\mu}{r^3}. \]
The figure represents a circuit in which a periodic E.M.F. \(V\cos pt\) is induced across \(EF\), and which contains between \(A\) and \(B\) a coil of resistance \(R\) and self-inductance \(L\). The resistance of the remainder of the circuit is \(r\). The points \(A\) and \(B\) are also connected by leads of total resistance \(r'\) to the plates of a condenser of capacity \(C\). Find the current in the main circuit.
Prove that \[ \left(\frac{dp}{dT}\right)_v = \frac{l_v}{T}, \] where \(p, v, T\) are respectively pressure, volume and temperature, and \(l_v\) is the latent heat of expansion at temperature \(T\). Defining a perfect gas as one in which \(pv\) and the internal energy are both functions of \(T\) only, show that the coefficient of cubical expansion at constant pressure is the same for all perfect gases at the same temperature. Show further that the difference of the specific heats is for the same perfect gas a constant.
Show that at a place in latitude \(\phi\) the duration of twilight is least when \[ \sin\delta = -\tan 9^\circ \sin\phi, \] \(\delta\) being the sun's declination, and, assuming that the sun moves uniformly in the ecliptic in 365 days, give a formula for the number of nights in which there is twilight all night. (Twilight lasts while the sun is less than 18\(^\circ\) below the horizon.)
Prove that the locus of the centre of a circle, which touches two given circles in a plane, consists of two hyperbolas, two ellipses or an ellipse and a hyperbola according as one given circle is external to, internal to or cuts the other.
Three lines in space do not intersect and are not all parallel to the same plane: prove that they are three edges of a parallelepiped and that one line can be drawn to intersect the three lines so that the intercept on it between a definite pair of the lines is bisected by the third line.
Prove that the rectangles contained by the segments of any two intersecting chords of a conic are to one another as the squares of the parallel tangents taken between their intersection and their points of contact. Two chords of an ellipse \(Pp, Qq\) are at right angles and are normal to the ellipse at \(P\) and \(Q\) respectively: show that \(pq\) is parallel to \(PQ\).
Two circles lie in different planes: prove that in general four circles can be drawn to touch both circles. What is the exceptional case in which an infinite number of such circles can be drawn?
Prove that a plane section of a circular cone is a conic section as defined by focus and directrix. Prove that through any point within the cone two sections can be drawn of which the point is a focus; show also that one of these sections is an ellipse and the other an ellipse, parabola or hyperbola according as the point is within, on or without another circular cone with the same vertex and axis.