A proton is moving in a region of uniform magnetic field. The magnetic field is directed into the plane of the paper. The arrow shows the velocity of the proton at one instant and the dotted circle gives the path

D$.$ End$-$of$-$theme questions

d$.$ i$.$ Show that the energy dissipated in the loop from $t=0$ to $t=3\mathrm{.}5$ s is $0\mathrm{.}13$ J$.$

ii$.$ The mass of the wire is $18g.$ The specific heat capacity of copper is $385Jk{g}^{-1}{k}^{-1}.$ Estimate the increase in temperature of the wire.

A conducting sphere has radius $48cm.$ The electric potential on the surface of the sphere is $3\mathrm{.}4{10}^{5}V.$

a$.$ Show that the charge on the surface of the sphere is $+18C.$

b$.$ The sphere is connected by a long conducting wire to a second conducting sphere of radius $24cm.$ The second sphere is initiallv uncharmad

a$.$ Explain why the path of the proton is a circle.

b$.$ The speed of the proton is $2\mathrm{.}0{10}^{6}m{s}^{-1}$ and the magnetic field strength $B$ is $0\mathrm{.}35T.$

i$.$ Show that the radius of the path is about $6cm.$

ii$.$ Calculate the time for one complete revolution.

c$.$ Explain why the kinetic energy of the proton is constant.

i$.$ Describe, in terms of electron flow, how the smaller sphere becomes charged.

ii$.$ Predict the charge on each sphere.

A square loop of side $5\mathrm{.}0cm$ enters a region of uniform magnetic field at $t=0.$ The loop exits the region of magnetic field at $t=3\mathrm{.}5s.$ The magnetic field strength is $0\mathrm{.}94T$ and is directed into the plane of the paper. The magnetic field extends over a length $65cm.$ The speed of the loop is constant.

a$.$ Show that the speed of the loop is $20cm{s}^{-1}.$

While the magnet is moving towards the ring:

b$.$ Sketch a graph to show the variation with time of:

a$.$ State why the magnetic flux in the ring is increasing.

i$.$ the magnetic flux linkage $$ in the loop.

b$.$ Sketch, using an arrow on Diagram $2,$ the direction of the induced current in the ring.

ii$.$ the magnitude of the emf induced in the loop.

c$.$ Deduce the direction of the magnetic force on the magnet.

c$.$ i$.$ There are $85$ turns of wire in the loop. Calculate the maximum induced emf in the loop.

A small magnet is dropped from rest above a stationary horizontal conducting ring. The south $($S$)$ pole of the magnet is upwards. of the induced current in the ring.

c$.$ Deduce the direction of the magnetic force on the magnet.