The magnetic field of the Earth can be approximated by that due to a dipole given by the expression in equation 2.1 in spherical polar coordinates: \vec{B}=\left(B_{r}, B_{\theta}, B_{\phi}\right)=\frac{A}{3}(-2 \cos \theta,-\sin \theta, 0) \text {. Equation } 2.1 Show that the magnitude of the field strength of this dipole can bewritten as |\vec{B}|=\frac{A}{r^{3}}\left(1+3 \cos ^{2} \theta\right)^{1 / 2} The latitude of Greenwich is 51.5 degrees. Assuming the magnetic pole and geographical pole are coincident, show that the polar angle for Greenwich is 38.5 degrees. A historical measurement of the magnetic field at Greenwich from 1846 [described by Barraclough et al. 2000, Geophys. J. Int. 141, 83-99] gives the following values for the magnitudes of the horizontal component and total field (in nano teslas): \text { Horizontal Field: } 17160 \mathrm{nT}

(i)Show that the vertical component of the field is given by44630 nT. (5 marks) (ii)Draw the field lines above the ground at Greenwich -indicate clearly the angle that the field lines make with the vertical. Do the field lines point in or out of the ground? Hint: The Earth's geographical north pole is a magnetic south pole. (6 marks) (iii)Do the measured values correspond to what you would expect for a pure dipole field at this location? Hint: insert your answer for (b) into equation 2.1, compute the ratio of the field components, and then compare with the ratio of the measured values. (8 marks) (iv)Assuming a pure dipole field for the Earth, and using the answer to part (b) and the total field value of 47815 nT,estimate a value for the constant A in Equation 2.1. You may take the Earth's radius at Greenwich to be 6371 km.(6 marks) The total magnetic energy stored in the magnetic field above the Earth's surface is given by the following integral in spherical polar coordinates: E=\frac{1}{2 \mu_{0}} \iiint B^{2} d V=\frac{1}{2 \mu_{0}} \iiint B^{2} r^{2} \sin \theta d r d \theta d \phi . (i)What are the limits of these three integrals appropriate for the energy stored above the Earth's surface? (6marks) (ii)Insert the expression from (a) for the magnitude of the field (together with the value for A from c- part iv) and do the three integrals. Hence calculate a value for this energy. (10 marks)