2021 A/L Physics Past Paper · Question 34 (English Medium)

Total force on a charge in parallel electric and magnetic fields

A uniform magnet field of flux density $(B)$ $1\ \text{T}$ and a uniform electric field of field intensity $(E)$ $300\ \text{V m}^{-1}$ are parallel to each other in a region as shown in the figure. A particle of charge $+q$ enters at a velocity $(v)$ $400\ \text{m s}^{-1}$ perpendicular to the fields and directed into the plane of the paper. The magnitude of the resultant force acting on the particle is

(1) $0$
(2) $100q$
(3) $300q$
(4) $500q$
(5) $700q$

Answer and explanation

Answer: (4) $500q$

What the question asks

We want: how big the total force on the charge is.

Two pushes act on the charge. One comes from the electric field. One comes from the magnetic field.

The plan: find each push on its own. Then put the two pushes together.

Relevant theory

Hence, the force acting on a charge "q" placed at a point of intensity E is F = Eq.

No force acts on an electric charge which is at rest in a magnetic field. However if the charge is moving it is equivalent to an electric current and hence a force would act on it.

Applying it to this question

The electric push

$F_E = 300q$ , along the field lines

An electric field pushes a charge along the field lines.

$F_E = qE$

In words: a bigger charge or a stronger field gives a bigger push.

$F_E = q \times 300 = 300q$ .

The magnetic push

$F_B = 400q$ , at right angles to the field lines

A moving charge feels a magnetic push.

$F_B = qvB\sin\theta$

In words: a faster charge, a stronger field or a bigger charge gives a bigger push.

The charge moves at $90^\circ$ to the field, so $\sin 90^\circ = 1$ .

$F_B = q \times 400 \times 1 \times 1 = 400q$ .

Every field line looks the same all along the region. So it does not matter where on the lines the charge is.

We can drop the position and use one value for each push.

Which way does each push go?

The electric push goes along the field lines.

The magnetic push is always at right angles to the magnetic field lines. The electric field lines run the same way as the magnetic ones.

So the magnetic push is at right angles to the electric push too.

Reasoning and calculation

Two pushes at a right angle, $300q$ and $400q$ . How big is the total?

Think of a walk. Go 3 steps east, then 4 steps north. How far are you from the start? You are 5 steps away, along a slanted line.

Squares add up: $3^2 + 4^2 = 5^2$ .

The two pushes are the same, just 100 times bigger.

$250000$

$300^2 + 400^2$

$500$

$\sqrt{250000}$

The total force is $500q$ .

Reading the options

The value $500q$ is option (4).

Option (5) adds the two pushes in a straight line. Option (2) takes one away from the other. Options (1) and (3) leave out a push.

Why the other options fail

Both pushes are there and they are at a right angle. They do not cancel.

Pushes at a right angle do not take away from each other.

A moving charge feels a magnetic push as well as the electric push.

The pushes are at a right angle, so they do not add in a straight line.

Checks

Does it make sense?

The total, $500q$ , is bigger than each push ( $300q$ and $400q$ ). It is smaller than the straight sum, $700q$ .

That is always so for two pushes at a right angle.

$sqrt(a * a + b * b)$

Conclusion

$F = \sqrt{F_E^2 + F_B^2} = 500q$

The electric push is $300q$ and the magnetic push is $400q$ . They are at a right angle.

The total is $\sqrt{300^2 + 400^2}\,q = 500q$ . The answer is (4).

Find each push on its own, and say which way it goes.

If two pushes are at a right angle, use the 3, 4, 5 walk. Do not add or subtract them in a straight line.

Exam tip

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