A plane mirror is placed behind a sphere of radius \(R\) and refractive index \(\mu\). Show that the effect on a small pencil which passes directly through the sphere, is reflected at the plane, and re-traverses the sphere, will be the same as if it had been reflected at a certain mirror. Find the situation and curvature of this mirror.
The resistances of the four sides of a Wheatstone's bridge are, in order, \(\alpha, \beta, b, a\). Prove that, whatever the resistance of the bridge connecting the junctions of \(\alpha, \beta\) and \(a,b\), the resistance of the whole framework lies between \[ \frac{(a+b)(\alpha+\beta)}{a+b+\alpha+\beta} \quad \text{and} \quad \frac{a\alpha}{a+\alpha} + \frac{b\beta}{b+\beta}. \]
An insulated spherical conductor \(C\) formed of two hemispherical shells in contact (of outer and inner radii \(b,c\)) is surrounded by a concentric hollow spherical conductor \(C_1\) of internal radius \(a\), and encloses a concentric spherical conductor \(C_2\) of radius \(d\). The potential of \(C_1\) is thrice that of \(C\), while \(C_2\) is at zero potential. Find the condition that the two hemispheres of \(C\) may just be held together electrically.
The radii of the inner and outer spheres of a spherical condenser are \(a,b\). The inner sphere is excentric by a small distance \(c\). The intervening space is filled with a dielectric of specific inductive capacity \(K\). The surface density at a point on the inner sphere at an angular distance \(\theta\) from the line of centres may be represented, to the first order in \(c\), by \(\lambda_1 V(1+\lambda_2 c\cos\theta)\), where \(V\) is the potential difference. Determine \(\lambda_1\) and \(\lambda_2\).
An infinite plane has a hemispherical boss upon it, the whole forming one conductor, which is put to earth in the presence of a point charge \(e\) at a point \(P\) along the axis of the boss. The angle subtended at \(P\) by any diameter of the boss which lies in the plane is \(2\theta\). Prove that the total charge on the boss is \(e(1-\cos 2\theta/\cos\theta)\), and also that the presence of the boss increases the force of attraction on the charge at \(P\), towards the plate, in the ratio \[ (1+2\sin 2\theta \cos^2\theta\tan^2 2\theta):1. \]
The "centre of mass," \(O\), of the electricity on a conductor, charged and alone in the field, is called the electric centre of the conductor. Prove that the potential at a point \(P\) in the field must lie between \[ \frac{E}{OP}\left(1+\frac{\sigma^2}{OP^2}\right) \quad \text{and} \quad \frac{E}{OP}\left(1-\frac{\sigma^2}{2OP^2}\right), \] where \(E\) is the total charge on the conductor, and \(\sigma\) is the greatest radius of the conductor from the electric centre \(O\). Also prove that if there are two conductors \(C, C'\) in the field, with electric centres \(O, O'\), and maximum radii \(a, a'\) measured from \(O, O'\), their mutual coefficient of potential is \(1/c'\), where \(c'\) cannot differ from \(OO'\) by more than \((a^2+a'^2)/OO'\).
Three infinite parallel wires cut a plane perpendicular to them in the angular points \(X,Y,O\) of an equilateral triangle, and have charges \(e,e,e_0\) per unit length respectively. Prove that the limiting lines of force which pass from \(X\) to \(O\) make at starting angles \((2e-5e_0)\pi/6e\), and \((2e+e_0)\pi/6e\) with \(XO\), provided that \(e>2e_0\). Sketch the lines of force, and determine the distance of the point of equilibrium from \(XY\).
Prove, by inversion or by the method of images, that if a small sphere, of radius \(a\), be made to touch a large one, of radius \(b\), the ratio of the mean density on the small to that on the large sphere tends to \(\pi^2/6\), as \(a/b \to 0\).
Two spheres, radii \(a,b\), have their centres at a distance \(c\) apart. Prove the approximate formula \(p_{12}=1/c\), showing that the error is of order \((a/c)^7\) or \((b/c)^7\).