Magnetic Fields

The potential difference (p.d.) between the plates of the velocity selector is $V$. The separation of the plates is $d$ and the magnetic flux density is $B$.

Show that the speed $u$ of ions that pass undeviated through the velocity selector is given by

Positive ions with kinetic energy $4.1 \times 10^{-17}\ \text{J}$ and mass $3.2 \times 10^{-27}\ \text{kg}$ pass undeviated through the velocity selector when $V$ is equal to $980\ \text{V}$ and $d$ is equal to $3.6 \times 10^{-2}\ \text{m}$.

Determine $B$.

$B$ = ______ $\text{T}$

A proton passes undeviated through the velocity selector.

An alpha particle enters the velocity selector at the same speed as the proton.

State how the expression in (a) predicts that the alpha particle also passes undeviated through the velocity selector.

By reference to Fig. 6.1 and to the forces acting on a positive ion, determine the direction of the magnetic field. Explain your reasoning.

The positive ions in (b) enter the velocity selector with greater kinetic energy.

On Fig. 6.1, sketch the path of these ions.

State Faraday’s law of electromagnetic induction.

Explain how Fig. 7.2 shows that $E$ is proportional to the velocity $v$ of the rod.

Use Faraday’s law to show that the variation of $E$ with time $t$ is given by

where $a$ is the acceleration of the rod.

The length of the rod is $0.45\ \text{m}$. The acceleration $a$ of the rod is $7.8\ \text{m s}^{-2}$.

Determine the value of $B$.

$B$ = ______ $\text{T}$

Define magnetic flux density.

State an expression, in terms of some or all of $m$, $Q$, $B$ and $v$, for the magnetic force $F$ that acts on the particle when it is at point Y.

$F$ = ______

On Fig. 7.1, draw an arrow at point Y to indicate the direction of the force in (b)(i).

On Fig. 7.1, draw a line to show a possible path for the particle through the region of the magnetic field.

Explain how an electric field can be used with the magnetic field to ensure that the particle in (b) now passes through point Z.

Derive an expression for $v$ in terms of $B$ and the electric field strength $E$.

$v$ = ______

Define magnetic flux density.

State an expression, in terms of some or all of $m$, $Q$, $B$ and $v$, for the magnetic force $F$ that acts on the particle when it is at point Y.

$F$ = ______

On Fig. 7.1, draw an arrow at point Y to indicate the direction of the force in (b)(i).

On Fig. 7.1, draw a line to show a possible path for the particle through the region of the magnetic field.

Explain how an electric field can be used with the magnetic field to ensure that the particle in (b) now passes through point Z.

Derive an expression for $v$ in terms of $B$ and the electric field strength $E$.

$v$ = ______

Calculate the magnitude of the force on side QR of the coil.

force = ______ $\text{N}$

Use your answer in (a)(i) to show that the torque $\tau$ on the coil is given by

Using the expression in (a)(ii) sketch, on the axes of Fig. 6.2, a graph to show the variation of the torque $\tau$ with angle $\theta$ for values of $\theta$ between $0$ and $360^{\circ}$. Label the $\tau$ axis with an appropriate scale.

The coil is now replaced by an identical coil wound on a ferrous core.

Suggest, with a reason, how the torque on this coil compares with the torque on the original coil.

State Faraday’s law of electromagnetic induction.

Calculate the magnetic flux cut by the wings of the aircraft in a time of $15\ \text{s}$. Give a unit with your answer.

magnetic flux = ______ unit ______

Determine the area of flux cut by the wings in a time of $15\ \text{s}$.

area = ______ $\text{m}^2$

Use your answer in (b)(ii) to determine the speed $v$ of the aircraft.

$v$ = ______ $\text{m s}^{-1}$

Use Lenz’s law of electromagnetic induction to explain which of the wingtips P and Q is at the higher induced potential.

State Faraday’s law of electromagnetic induction.

Calculate the magnetic flux cut by the wings of the aircraft in a time of $15\ \text{s}$. Give a unit with your answer.

magnetic flux = ______ unit ______

Determine the area of flux cut by the wings in a time of $15\ \text{s}$.

area = ______ $\text{m}^2$

Use your answer in (b)(ii) to determine the speed $v$ of the aircraft.

$v$ = ______ $\text{m s}^{-1}$

Use Lenz’s law of electromagnetic induction to explain which of the wingtips P and Q is at the higher induced potential.

State Lenz’s law of electromagnetic induction.

Calculate the magnetic flux $\Phi$ cut by rotor OX during one complete rotation. Give a unit with your answer.

$\Phi$ = ______ unit ______

Determine the magnitude of the electromotive force (e.m.f.) induced across the length of rotor OX.

e.m.f. = ______ $\text{V}$

Use Lenz’s law to explain whether end O or end X of the rotor is at the higher potential.

There is a constant current in the solenoid.
Coil C is moved through the solenoid from position X to position Y.

On Fig. 6.2, sketch a line to show the variation of the magnetic flux linkage in coil C with position as it moves from X to Y.

Explain the shape of your line in (a)(i).

Coil C is now held stationary at X. The current in the solenoid varies so that the magnetic flux density $B$ at X varies from time 0 to time $4t$ as shown in Fig. 6.3.

Calculate the maximum magnetic flux linkage in coil C.

flux linkage = ______ $\text{Wb}$

On Fig. 6.4, sketch a line to show the induced electromotive force (e.m.f.) $E$ in coil C from time 0 to time $4t$.

A metal spring rests on a smooth table. The turns of the spring are equally spaced. The ends of the spring are connected to a d.c. power supply, as shown in Fig. 6.5.

The spring is connected to the d.c. power supply using flexible leads. The spring is not under tension.

With reference to magnetic fields, describe and explain the change in the distance between the turns of the spring when the power supply is first switched on.

State Faraday’s law of electromagnetic induction.

Calculate the maximum magnetic flux through one turn of the coil.

maximum magnetic flux = ______ $\text{Wb}$

Determine the maximum rate of change of magnetic flux linkage in the coil.

maximum rate of change of flux linkage = ______ $\text{Wb s}^{-1}$

State the maximum electromotive force (e.m.f.) $V_0$ induced across the coil.

$V_0$ = ______ $\text{V}$

On Fig. 7.3, sketch the variation of the e.m.f. $V$ induced across the coil with $t$ from $t = 0$ to $t = 6.0\ \text{ms}$.

The variation of $V$ with $t$ can be described by

where $A$ and $B$ are constants.

Determine the values of $A$ and $B$. Give units with your answers.

$A$ = ______ unit ______
$B$ = ______ unit ______