How to Draw a Compass Needle Near a Current-Carrying Wire

Put a straight wire directly above a compass, with both aligned north-south. Close the switch so conventional current flows north along the wire. The magnetic field produced by the wire at the compass points west. Earth’s horizontal magnetic field still points north, so the compass’s north-seeking end turns towards the north-west.

The important detail is that the compass does not simply point in the direction of the current or even along the magnetic field produced by the wire alone. It aligns with the resultant magnetic field, which combines the wire’s field with Earth’s magnetic field.

To draw the needle correctly, you need three things: the right-hand grip rule, the position of the compass relative to the wire, and the direction of the resultant field.

01What is the experiment actually testing?

The experiment addresses a simple question: can an electric current affect a nearby magnet?

A compass needle is a small permanent magnet. Before the current flows, its north-seeking end aligns approximately with Earth’s magnetic field.

Now place a straight conducting wire directly above the compass. Keep the wire parallel to the compass needle, with both pointing north-south.

Connect the wire to a DC power supply through a switch and an ammeter. The ammeter measures the current flowing through the wire.

Predict what happens

Suppose the switch is open. No current flows through the test wire, so the compass responds primarily to Earth’s magnetic field.

Its north-seeking end points towards magnetic north.

What should happen when the switch closes?

If the current has no magnetic effect, the needle should remain pointing north. However, when current flows, the needle deflects.

Reversing the current reverses the direction of the deflection.

This observation shows that an electric current produces a magnetic field around the wire.

The experiment is especially useful because the compass makes an otherwise invisible magnetic field observable. It does not directly show the complete field pattern, though. It shows the direction of the resultant field at the compass’s position.

Why the apparatus arrangement matters

There is an important experimental limitation.

If the return wire from the power supply passes close to the compass, it produces its own magnetic field. The compass responds to both wires, so the measured deflection may not represent the effect of the straight test section alone.

A better arrangement keeps the return lead well away from the compass, uses a sufficiently long straight test section, and maintains a fixed wire-to-compass separation.

You can also compare three observations:

Reversing the current while keeping the rest of the apparatus unchanged is particularly useful. It shows that the direction of the magnetic effect depends on the direction of current.

One more detail: the compass must be positioned correctly. If the horizontal wire runs north-south and the compass sits directly beside it at the same height, the wire’s magnetic field at that position is vertical. A horizontal compass will not show the same east-west deflection as it does when placed directly above or below the wire.

02How to draw the compass needle: five steps

Use this method whenever an HSC question asks you to draw or predict the orientation of a compass near a current-carrying wire.

For the standard arrangement, assume:

• The wire is horizontal and runs north-south.
• The compass is directly below the wire.
• The compass and wire are separated by a perpendicular distance \(r\).
• The current is steady.
• Earth’s horizontal magnetic field points north.
• Other magnetic fields are negligible.

Here, north means the direction an undisturbed compass points. We will ignore magnetic declination.

Step 1: Identify the wire’s position and direction

First, establish where the compass is relative to the wire.

This is essential because the magnetic field reverses direction when you move from one side of a wire to the other.

For our standard arrangement, the wire is above the compass.

Step 2: Mark conventional current

Draw an arrow showing the direction of conventional current.

Conventional current flows in the direction positive charge would move, from the positive terminal towards the negative terminal through the external circuit.

In a metal wire, electrons drift in the opposite direction. Do not use electron flow as the direction of your right-hand-rule thumb.

Step 3: Find the magnetic field at the compass

Use the right-hand grip rule:

1. Point your right thumb in the direction of conventional current.
2. Curl your fingers around the wire.
3. Your curled fingers show the direction of the magnetic field.

For the standard arrangement:

• Current north: magnetic field at the compass points west.
• Current south: magnetic field at the compass points east.

Notice that the magnetic field is perpendicular to the wire, not parallel to the current.

If you need to rebuild the circular field pattern at other positions, see Magnetic Field Around a Current-Carrying Wire: HSC Physics.

Step 4: Add Earth’s magnetic field

This is the step students most often miss.

The compass is not responding to the wire’s magnetic field in isolation. Earth’s magnetic field is still present.

There are therefore two magnetic fields to consider:

• The wire’s field, \(B_\mathrm{wire}\), pointing west.
• Earth’s horizontal field, \(B_H\), pointing north.

Draw both arrows from the same starting point.

The resultant field points between north and west.

Step 5: Draw the compass needle along the resultant

The north-seeking end of the compass needle points in the direction of the resultant magnetic field.

The south-seeking end points in the opposite direction.

For our standard arrangement:

• North end: towards the north-west.
• South end: towards the south-east.

The exact angle depends on the relative strengths of the two fields.

If Earth’s field is much stronger, the needle stays close to north. If the wire’s field is much stronger, the needle turns closer to west.

Important: If a question explicitly asks you to ignore Earth’s magnetic field, you can draw the needle along the wire’s magnetic field alone. Otherwise, include Earth’s field.

03Magnetic field rules versus magnetic force rules

There are two different ideas that are easy to confuse.

The right-hand grip rule determines the direction of the magnetic field produced by a current. A motor-effect direction rule determines the force on a current-carrying conductor in an external magnetic field.

They answer different questions.

For the motor effect, the force magnitude is given by

\[
F=BIL\sin\theta
\]

where \(F\) is force in newtons, \(B\) is magnetic flux density in tesla, \(I\) is current in amperes, \(L\) is conductor length in metres, and \(\theta\) is the angle between the current and the magnetic field.

You do not use this force equation to determine which way a compass needle points.

The compass is a magnet. The magnetic field exerts a turning effect on it, causing the needle to align with the resultant field.

If a question asks how two current-carrying wires push or pull each other, that is a different application of magnetic fields. See Force Between Parallel Current-Carrying Wires: HSC Physics.

04Calculating the compass deflection

So far, we have determined the direction of the needle. But how far does it turn?

For a long, straight current-carrying wire, the magnetic flux density at perpendicular distance \(r\) is

\[
B_\mathrm{wire}=\frac{\mu_0 I}{2\pi r}
\]

where:

• \(B_\mathrm{wire}\) is the magnetic flux density produced by the wire, in tesla (\(\text{T}\)).
• \(\mu_0\) is the permeability of free space, approximately \(4\pi\times10^{-7}\ \text{T m A}^{-1}\).
• \(I\) is conventional current, in amperes (\(\text{A}\)).
• \(r\) is the perpendicular distance from the wire, in metres (\(\text{m}\)).

This equation assumes a sufficiently long, straight wire and is a good approximation for a wire in air.

For our standard arrangement, the wire’s magnetic field is perpendicular to Earth’s horizontal magnetic field.

We can therefore use a right-angled vector triangle.

Let \(\theta\) be the angle the compass needle turns away from magnetic north.

Then

\[
\tan\theta=\frac{B_\mathrm{wire}}{B_H}
\]

This gives us a useful result:

• Increasing current increases the deflection.
• Moving the compass closer to the wire increases the deflection.
• Increasing Earth’s horizontal field decreases the deflection.

The equation also shows why the compass does not automatically point due west. Both magnetic fields contribute to its final orientation.

05What changes when you reverse the current?

Consider the same arrangement, but reverse the current so that it flows south.

The right-hand grip rule now gives an eastward magnetic field at the compass.

Earth’s magnetic field has not changed. It still points north.

The resultant field therefore points towards the north-east.

Notice something important: reversing the current does not necessarily reverse the compass needle by 180°.

Only the wire’s magnetic field reverses. Earth’s magnetic field remains unchanged, so the resultant generally points into the opposite side of north.

The side of the wire matters too

There is another possible change students often overlook.

Moving the compass from below the wire to above it also reverses the direction of the wire’s magnetic field at the compass.

For a horizontal north-south wire:

This gives you a useful diagnostic rule:

The magnetic field direction at the compass depends on both the current direction and which side of the wire the compass occupies.

Reversing the current and moving the compass to the opposite side are two separate changes. If both happen together, their effects on the wire’s field cancel, leaving its original direction unchanged.

06Questions and solutions

The following questions progress from basic direction rules to more demanding situations involving field strength, changing geometry, and interpreting experimental observations.

Question 1

A long straight horizontal wire runs north-south directly above a compass. The conventional current flows south along the wire.

Earth’s horizontal magnetic field points north.

Determine:

a. The direction of the magnetic field produced by the wire at the compass.

b. The direction in which the north-seeking end of the compass needle deflects.

Explain your reasoning.

Solution 1

The wire’s magnetic field points east, and the compass’s north-seeking end deflects east of magnetic north.

Step 1: Apply the right-hand grip rule.

Point your right thumb south, along the conventional current.

Your fingers show the direction of the circular magnetic field.

At the position directly below the wire, the field points east.

Step 2: Include Earth’s magnetic field.

Earth’s horizontal field points north.

The wire’s field points east.

These two perpendicular fields combine to produce a resultant towards the north-east.

Step 3: Draw the compass.

The north-seeking end points towards the north-east, while the south-seeking end points towards the south-west.

The needle does not necessarily point exactly 45° east of north. That would require the two field strengths to be equal.

The important distinction is that the wire determines the direction of its own field, while both fields determine the final compass orientation.

Question 2

A long straight wire is positioned \(5.0\ \text{cm}\) directly above a compass. Both are aligned north-south.

The wire carries a conventional current of \(4.0\ \text{A}\) northwards.

The horizontal component of Earth’s magnetic field is \(2.0\times10^{-5}\ \text{T}\) north.

Calculate the angle of deflection of the compass needle from magnetic north.

Solution 2

The compass deflects approximately \(39^\circ\) west of magnetic north.

Step 1: Determine the direction of the wire’s magnetic field.

The current flows north, and the compass is below the wire.

The right-hand grip rule gives a magnetic field pointing west at the compass.

Step 2: Convert the separation into metres.

\[
r=5.0\ \text{cm}=0.050\ \text{m}
\]

Step 3: Calculate the magnetic field produced by the wire.

\[
B_\mathrm{wire}=\frac{\mu_0 I}{2\pi r}
\]

Using \(\mu_0/(2\pi)=2.0\times10^{-7}\),

\[
\begin{aligned}
B_\mathrm{wire}
&=(2.0\times10^{-7})
\frac{4.0}{0.050}\\
&=1.6\times10^{-5}\ \text{T}
\end{aligned}
\]

Step 4: Calculate the deflection angle.

\[
\tan\theta=\frac{B_\mathrm{wire}}{B_H}
\]

\[
\begin{aligned}
\tan\theta
&=\frac{1.6\times10^{-5}}
{2.0\times10^{-5}}\\
&=0.80
\end{aligned}
\]

Therefore,

\[
\theta=\tan^{-1}(0.80)
\]

\[
\theta=38.7^\circ
\]

Step 5: State the final answer.

\[
\boxed{\theta=39^\circ\text{ west of north}}
\]

The wire’s field is slightly weaker than Earth’s horizontal field, so the compass turns towards west without pointing directly west.

Question 3

A long straight wire runs north-south, directly above a compass. The separation between the wire and compass is \(2.5\ \text{cm}\).

Earth’s horizontal magnetic field is \(2.0\times10^{-5}\ \text{T}\) north.

When the switch is closed, the compass’s north-seeking end deflects \(30^\circ\) east of magnetic north.

Determine:

a. The magnitude of the magnetic field produced by the wire at the compass.

b. The magnitude and conventional direction of the current in the wire.

Solution 3

The current is approximately \(1.4\ \text{A}\) southwards.

Step 1: Determine the direction of the wire’s magnetic field.

The compass deflects east of north.

Earth’s field points north, so the wire must produce an eastward magnetic field.

Because the compass is below the wire, the right-hand grip rule tells us that the conventional current must flow south.

Step 2: Calculate the wire’s magnetic field.

For perpendicular fields,

\[
\tan\theta=\frac{B_\mathrm{wire}}{B_H}
\]

Rearranging,

\[
B_\mathrm{wire}=B_H\tan\theta
\]

Substitute:

\[
\begin{aligned}
B_\mathrm{wire}
&=(2.0\times10^{-5})\tan30^\circ\\
&=(2.0\times10^{-5})(0.577)\\
&=1.15\times10^{-5}\ \text{T}
\end{aligned}
\]

Therefore, the wire produces a magnetic field of approximately \(1.2\times10^{-5}\ \text{T}\).

Step 3: Calculate the current.

Start with the magnetic field equation:

\[
B_\mathrm{wire}=\frac{\mu_0 I}{2\pi r}
\]

Rearrange to make current the subject:

\[
I=\frac{2\pi rB_\mathrm{wire}}{\mu_0}
\]

Substitute:

\[
\begin{aligned}
I
&=\frac{(2\pi)(0.025)(1.15\times10^{-5})}
{4\pi\times10^{-7}}\\
&=1.44\ \text{A}
\end{aligned}
\]

Step 4: State the complete answer.

\[
\boxed{I=1.4\ \text{A southwards}}
\]

The calculation determines the current’s magnitude. The right-hand grip rule determines its direction.

This distinction is important when working backwards from a compass reading: the angle tells you the relative field strengths, while the direction of deflection tells you which way the wire’s field points.

Question 4

A straight horizontal wire runs east-west and is positioned \(4.0\ \text{cm}\) directly above a compass.

The compass is in a region where Earth’s horizontal magnetic field is \(2.0\times10^{-5}\ \text{T}\) north.

Initially, the wire carries a conventional current of \(4.0\ \text{A}\) eastwards.

Answer the following:

a. Determine the direction and magnitude of the resultant magnetic field at the compass.

b. The current is reversed and increased to \(6.0\ \text{A}\) westwards. Determine the new resultant field and the compass’s direction.

c. At what magnitude of westward current would the horizontal resultant field become zero?

Solution 4

Initially, the resultant field is \(4.0\times10^{-5}\ \text{T}\) north. After the current is reversed and increased to \(6.0\ \text{A}\), the resultant is \(1.0\times10^{-5}\ \text{T}\) south. The north-seeking end of the compass then points south.

Step 1: Find the initial direction of the wire’s field.

The current flows eastwards, and the compass is below the wire.

Applying the right-hand grip rule gives a magnetic field pointing north at the compass.

This is an important change from our earlier examples. The wire now runs east-west rather than north-south.

Step 2: Calculate the initial wire field.

\[
r=0.040\ \text{m}
\]

\[
\begin{aligned}
B_\mathrm{wire}
&=(2.0\times10^{-7})\frac{4.0}{0.040}\\
&=2.0\times10^{-5}\ \text{T}
\end{aligned}
\]

Both Earth’s field and the wire’s field point north.

Therefore, they add directly:

\[
\begin{aligned}
B_\mathrm{net}
&=B_H+B_\mathrm{wire}\\
&=2.0\times10^{-5}
+2.0\times10^{-5}\\
&=4.0\times10^{-5}\ \text{T}
\end{aligned}
\]

The compass remains pointing north, even though the magnetic field has doubled in magnitude.

Step 3: Reverse the current and increase its magnitude.

The new conventional current is \(6.0\ \text{A}\) westwards.

Reversing the current reverses the wire’s magnetic field, so it now points south.

Calculate its magnitude:

\[
\begin{aligned}
B_\mathrm{wire}
&=(2.0\times10^{-7})\frac{6.0}{0.040}\\
&=3.0\times10^{-5}\ \text{T}
\end{aligned}
\]

The wire’s field is now stronger than Earth’s horizontal field.

The two fields point in opposite directions, so subtract their magnitudes:

\[
\begin{aligned}
B_\mathrm{net}
&=3.0\times10^{-5}
-2.0\times10^{-5}\\
&=1.0\times10^{-5}\ \text{T}
\end{aligned}
\]

The resultant points south.

Therefore, the compass’s north-seeking end points south.

Step 4: Find the current required for zero resultant field.

The horizontal fields cancel when their magnitudes are equal.

\[
B_\mathrm{wire}=B_H
\]

Using the wire field equation,

\[
2.0\times10^{-5}
=(2.0\times10^{-7})\frac{I}{0.040}
\]

Rearranging,

\[
\begin{aligned}
I
&=\frac{(2.0\times10^{-5})(0.040)}
{2.0\times10^{-7}}\\
&=4.0\ \text{A}
\end{aligned}
\]

Thus, a westward current of \(4.0\ \text{A}\) produces exact cancellation of the horizontal fields.

At this point, the compass has no preferred horizontal direction in the idealised model. In a real experiment, small background fields and other disturbances would determine its orientation.

The interesting result is that the compass can point north, become directionally unstable, and then point south as the current is increased. The direction of the wire’s field alone does not tell you the final compass direction. You must compare both field strengths.

Question 5

A long straight horizontal wire runs north-south. A compass is positioned \(3.0\ \text{cm}\) from the wire, and a conventional current of \(3.0\ \text{A}\) flows through it.

An experimental record shows that the compass’s north-seeking end points \(45^\circ\) west of magnetic north.

However, the record does not say whether the compass was above or below the wire.

Earth’s horizontal magnetic field is \(2.0\times10^{-5}\ \text{T}\) north.

Assume the wire is sufficiently long, the compass is directly above or below it, and other magnetic fields are negligible.

a. Determine whether the measured deflection is consistent with the given current and separation.

b. Identify all possible combinations of compass position and conventional current direction that could produce the observation.

c. The compass is moved to the opposite side of the same wire, maintaining the same perpendicular separation. The current is not changed. Predict the new compass bearing.

Explain why the original observation cannot uniquely determine the current direction.

Solution 5

The measurements are consistent, but the current direction cannot be uniquely determined without knowing which side of the wire contains the compass.

The two possible arrangements are:

• Compass below the wire, with current flowing north.
• Compass above the wire, with current flowing south.

After moving the compass to the opposite side without reversing the current, its north-seeking end points \(45^\circ\) east of magnetic north.

Step 1: Calculate the magnetic field produced by the wire.

The perpendicular separation is

\[
r=3.0\ \text{cm}=0.030\ \text{m}
\]

Using the long straight wire equation,

\[
B_\mathrm{wire}=\frac{\mu_0 I}{2\pi r}
\]

Substitute:

\[
\begin{aligned}
B_\mathrm{wire}
&=(2.0\times10^{-7})
\frac{3.0}{0.030}\\
&=2.0\times10^{-5}\ \text{T}
\end{aligned}
\]

The wire produces a magnetic field of \(2.0\times10^{-5}\ \text{T}\).

Step 2: Check the measured compass angle.

The Earth’s horizontal field is also

\[
B_H=2.0\times10^{-5}\ \text{T}
\]

The two fields are perpendicular in this arrangement.

Therefore,

\[
\tan\theta=\frac{B_\mathrm{wire}}{B_H}
\]

\[
\tan\theta=\frac{2.0\times10^{-5}}{2.0\times10^{-5}}=1
\]

Hence,

\[
\theta=45^\circ
\]

The calculated deflection matches the experimental observation.

The data are internally consistent.

Step 3: Identify the possible arrangements.

We know the wire’s magnetic field must point west because the compass deflects west of north.

However, there are two ways to produce a westward field.

First possibility:

The compass is below the wire.

If conventional current flows north, the right-hand grip rule gives a westward magnetic field at the compass.

Second possibility:

The compass is above the wire.

If conventional current flows south, the right-hand grip rule again gives a westward magnetic field at the compass.

Both arrangements produce exactly the same magnetic field at the compass.

Step 4: Move the compass to the opposite side.

Now keep the current unchanged and move the compass to the opposite side of the wire.

The magnetic field produced by the wire reverses direction at the new position.

It now points east rather than west.

Its magnitude remains unchanged because the perpendicular distance and current magnitude are unchanged.

Earth’s horizontal field still points north.

The two fields therefore combine to produce a resultant at

\[
45^\circ\text{ east of magnetic north}
\]

Step 5: Explain the ambiguity.

The tempting assumption is that a compass pointing west of north must indicate a particular current direction.

That conclusion is incomplete.

The direction of the magnetic field at a point depends on both:

• The direction of conventional current.
• The position of that point relative to the wire.

Reversing the current reverses the field. Moving to the opposite side of the wire also reverses the field.

When both changes happen together, the original field direction is restored.

This is why a compass reading alone cannot always reveal the direction of current. You must know the geometry of the experiment as well.

07Where this leads next

Drawing a compass needle near a current-carrying wire is an application of magnetic field superposition: the total field at a point is the vector sum of all contributing magnetic fields.

Once you can determine the resultant field for one wire, you can apply the same reasoning to multiple conductors, current loops, and solenoids.

The key habit is to keep three questions separate:

1. Which way does conventional current flow?
2. Which way does that current produce a magnetic field at the compass?
3. Which way does the resultant field point after Earth’s magnetic field is included?

Get those three steps in the right order, and you can draw the compass needle without confusing magnetic field direction with magnetic force direction.