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A cyclotron has two D-shaped regions where the magnetic flux density is constant - AQA - A-Level Physics - Question 3 - 2017 - Paper 2

Question 3

A cyclotron has two D-shaped regions where the magnetic flux density is constant. The D-shaped regions are separated by a small gap. An alternating electric field be... show full transcript

Worked Solution & Example Answer:A cyclotron has two D-shaped regions where the magnetic flux density is constant - AQA - A-Level Physics - Question 3 - 2017 - Paper 2

Step 1

Explain why it is not possible for the magnetic field to alter the speed of a proton while it is in one of the D-shaped regions.

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Answer

The magnetic field does not work on the proton as the magnetic force is always perpendicular to its velocity. This means there is no component of force in the direction of motion, and thus, the speed of the proton remains unchanged.

Step 2

Derive an expression to show that the time taken by a proton to travel round one semi-circular path is independent of the radius of the path.

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Answer

To derive the expression, we start with the physics of circular motion. The centripetal force required to keep the proton in a circular path is provided by the magnetic force:

$F = qvB$

where:

$F$ is the centripetal force,
$q$ is the charge of the proton,
$v$ is the speed of the proton,
$B$ is the magnetic flux density.

For circular motion, we have:

$F = \frac{mv^2}{r}$

Equating the two forces gives:

$qvB = \frac{mv^2}{r}$

From this, rearranging to find $t$ , the time for one full semi-circular path (half a full circle) yields:

$t = \frac{2\pi r}{v}$

Since $v$ can be expressed from the original equations, we find that $t$ ultimately does not depend on $r$ .

Step 3

Calculate the maximum speed of a proton when it leaves the cyclotron.

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Answer

To calculate the maximum speed, we use the given values for the magnetic flux density and the radius. The expression for speed derived is:

$v = \frac{qBr}{m}$

Using the following values:

$q = 1.6 \times 10^{-19}\, C$ (charge of the proton),
$B = 0.44\, T$ ,
$r = 0.55\, m$ , and,
$m = 1.67 \times 10^{-27}\, kg$ (mass of the proton).

Inserting these into the equation gives:

$v = \frac{(1.6 \times 10^{-19}) (0.44) (0.55)}{(1.67 \times 10^{-27})}\approx 2.43 \times 10^{7}\, m/s$

Thus, the maximum speed of the proton when it leaves the cyclotron is approximately $2.43 \times 10^{7}\, m/s$ .

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A cyclotron has two D-shaped regions where the magnetic flux density is constant - AQA - A-Level Physics - Question 3 - 2017 - Paper 2

Worked Solution & Example Answer:A cyclotron has two D-shaped regions where the magnetic flux density is constant - AQA - A-Level Physics - Question 3 - 2017 - Paper 2

Explain why it is not possible for the magnetic field to alter the speed of a proton while it is in one of the D-shaped regions.

Derive an expression to show that the time taken by a proton to travel round one semi-circular path is independent of the radius of the path.

Calculate the maximum speed of a proton when it leaves the cyclotron.

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