D. Gravitational, Electric, and Magnetic Fields · Gravitational, Electric and Magnetic FieldsMagnetic Fields of Currents and the Force on a Conductor
D2.4 D3.3 D2.1About 75 minutes
Learning goals- I can describe and draw the magnetic fields of a straight conductor and a solenoid, and compare them with electric and gravitational fields.
- I can calculate magnetic field strength near a long straight wire (B = μ₀I/2πr) and inside a solenoid (B = μ₀NI/L).
- I can use F = ILB sin θ and the right-hand rule to find the force on a current-carrying conductor.
- I can explain why parallel currents attract or repel and calculate the force between them.
Words to know
magnetic field (B)
The field around magnets and moving charges. Its strength is measured in teslas (T). Its direction is the way a compass's north pole points.
tesla (T)
The SI unit of magnetic field strength: 1 T = 1 N/(A·m). Earth's field is about 5 × 10⁻⁵ T; a hospital MRI magnet is typically 1.5 T or 3 T.
permeability of free space (μ₀)
μ₀ = 4π × 10⁻⁷ T·m/A, the constant in the formulas for magnetic fields made by currents.
solenoid
A long coil of wire. A current in it makes a strong, nearly uniform field inside and a field like a bar magnet's outside.
motor principle
A current-carrying conductor in an external magnetic field feels a force perpendicular to both the current and the field.
From motors to magnitudes
Remember from Grade 11 (SPH3U, Electricity and Magnetism): a current makes a magnetic field (Oersted's principle); you found its direction with right-hand rule #1 (grip a wire, thumb along the conventional current, fingers curl with the field) and right-hand rule #2 (fingers curl with the current in a coil, thumb points to the N end). The motor principle said a current-carrying wire in a magnetic field feels a force, with right-hand rule #3: thumb along the current, straight fingers along the field, palm pushes in the direction of the force. With electron flow, use the left hand.
That unit ended with a promise: in Grade 12 you would calculate the size of these fields and forces. Here it is. (This lesson uses conventional current and right hands throughout.)
Measuring magnetic fields: the tesla
Magnetic field strength is B, measured in teslas (T). Some typical values:
- Earth's field at the surface: about 5 × 10⁻⁵ T (around 55 μT in southern Ontario).
- A fridge magnet near its surface: a few thousandths of a tesla.
- A hospital MRI magnet: usually 1.5 T or 3 T.
- The strongest steady fields made in laboratories: several tens of teslas.
Magnetic field lines differ from electric and gravitational field lines in one big way: they have no start or end. They form closed loops, coming out of a magnet's N pole, going around to the S pole and continuing through the inside of the magnet. That is because there are no isolated magnetic "charges" (no one has ever found a single N pole without an S pole).
Fields made by currents
Long straight wire. The field lines are circles around the wire (RHR #1). The field strength at distance r is
B = μ₀I/(2πr) with μ₀ = 4π × 10⁻⁷ T·m/A
So B is proportional to the current and inversely proportional to r (not r²: a wire is a line, not a point).
Solenoid. Inside a long coil with N turns over length L, the field is strong, uniform and along the axis (RHR #2):
B = μ₀NI/L
It does not depend on the radius of the coil or on where you are inside it (away from the ends). Outside, the solenoid's field looks just like a bar magnet's.
These two cases are the magnetic version of the field diagrams you drew earlier: the straight wire's circles are like a point charge's radial lines (strongest close in), while the inside of a solenoid is the magnetic version of the uniform field between parallel plates.
Worked exampleExample 1: Field near a wire
A long straight wire carries 5.0 A. Find the magnetic field 2.0 cm from it and compare with Earth's field.
- B = μ₀I/(2πr) = (4π × 10⁻⁷)(5.0)/(2π × 0.020).
- Cancel π: B = (2 × 10⁻⁷)(5.0)/0.020 = 5.0 × 10⁻⁵ T.
5.0 × 10⁻⁵ T, about the same as Earth's field. That is why Oersted's compass twitched near a wire.
Worked exampleExample 2: Field inside a solenoid
A solenoid 20 cm long has 500 turns and carries 2.0 A. Find the field inside.
- B = μ₀NI/L = (4π × 10⁻⁷)(500)(2.0)/0.20.
- B = (1.2566 × 10⁻⁶)(1000)/0.20 = 6.3 × 10⁻³ T.
6.3 × 10⁻³ T (6.3 mT), more than 100 times Earth's field, and uniform inside the coil. An iron core would make it much stronger.
The force on a conductor: F = ILB sin θ
A straight wire of length L carrying current I in a uniform field B feels a force of size
F = ILB sin θ
where θ is the angle between the wire (current direction) and the field.
- Maximum force when the wire is perpendicular to B (θ = 90°): F = ILB.
- No force when the wire is parallel to B (θ = 0°).
- Direction: perpendicular to both I and B, given by right-hand rule #3.
This formula also defines the tesla: a field of 1 T exerts 1 N on 1 m of wire carrying 1 A at right angles, so 1 T = 1 N/(A·m).
Worked exampleExample 3: Force on a wire
A 25 cm length of wire carries 8.0 A in a uniform 0.40 T field. Find the force when the wire is (a) perpendicular to the field and (b) at 30° to the field.
- (a) F = ILB sin 90° = (8.0)(0.25)(0.40)(1) = 0.80 N.
- (b) F = (8.0)(0.25)(0.40) sin 30° = 0.80 × 0.50 = 0.40 N.
(a) 0.80 N; (b) 0.40 N. In both cases the force is perpendicular to the plane containing the wire and the field.
Worked exampleExample 4: Floating a wire
A horizontal copper rod with a mass of 0.020 kg per metre of length runs east-west in a horizontal magnetic field of 0.50 T pointing north. What current, in which direction, would make the magnetic force hold the rod up against gravity?
- For a length L: ILB = mg, so I = (m/L)g/B = (0.020)(9.8)/0.50 = 0.39 A.
- Direction: we need F up with B north. RHR #3: fingers north, palm up: the thumb points east.
0.39 A, flowing east. (Check: thumb east, fingers north, palm faces up.)
Parallel currents
Two parallel wires each make a field at the other, so each feels a force. For wires a distance d apart, each of length L:
F/L = μ₀I₁I₂/(2πd)
- Currents in the same direction attract.
- Currents in opposite directions repel.
(Notice this is the opposite of what you might guess from "like charges repel".) Until 2019, this force was used to define the ampere: 1 A was the current that gave a force of exactly 2 × 10⁻⁷ N per metre between two long wires 1 m apart. Today the ampere is defined from the fixed value of e, but the force is still real: the conductors in large electrical installations must be braced against it when currents surge.
Just for funA fridge magnet's field near its surface is a few thousandths of a tesla, roughly a hundred times Earth's. An MRI scanner's 3 T is about a thousand times stronger than the fridge magnet. Your grocery list would not stand a chance.
Hands-on activityMeasure a magnetic field with your phone
Most smartphones have a magnetometer (used for the compass app). Free apps such as phyphox (from RWTH Aachen University) show its reading in microteslas (μT).
You need: a phone with a magnetometer app, a D-cell or AA battery, about 1 m of insulated wire, tape, a ruler. A grown-up should check the setup: connect the wire across the battery only for a few seconds at a time, because the wire and battery get warm.
- Lay the phone flat, away from metal. Record the reading with no current (this is mostly Earth's field, about 50 to 60 μT).
- Tape the wire in a straight line across the table, passing over the spot where the magnetometer sits (often near the top of the phone; experiment to find it). Raise it on a book so it is 1 cm, then 2 cm, then 4 cm above the sensor.
- At each height, briefly touch the wire ends to the battery terminals and record the change in reading. Reverse the battery and repeat.
- Plot the change in B against 1/r.
What to look for: the change reverses when the current reverses, and it roughly halves each time the distance doubles (B ∝ 1/r). A short circuit across a D cell can carry a few amperes, so expect changes of tens of μT at 1 cm: comparable to Earth's field. Never use a wall outlet or a car battery for this.
Your turn
1Find the magnetic field 4.0 cm from a long wire carrying 12 A.
B = (2 × 10⁻⁷)(12)/0.040 = 6.0 × 10⁻⁵ T.
2A 0.50 m wire carrying 3.0 A is perpendicular to a 0.20 T field. Find the force.
F = ILB = (3.0)(0.50)(0.20) = 0.30 N.
3A 0.10 m wire carrying 5.0 A feels a 0.15 N force when perpendicular to a uniform field. Find B.
B = F/(IL) = 0.15/(5.0 × 0.10) = 0.30 T.
4A 0.20 m wire carrying 4.0 A in a 0.50 T field feels a 0.20 N force. Find the angle between the wire and the field.
sin θ = F/(ILB) = 0.20/(4.0 × 0.20 × 0.50) = 0.50, so θ = 30° (or 150°).
5A wire carries current north through a field that points straight down. Which way is the force?
RHR #3: thumb north, fingers down; the palm faces west. The force is west.
6Two parallel wires 5.0 cm apart carry 10 A and 15 A in the same direction. Find the force on a 2.0 m length of either wire. Attract or repel?
F = μ₀I₁I₂L/(2πd) = (2 × 10⁻⁷)(10)(15)(2.0)/0.050 = 1.2 × 10⁻³ N, attractive (same direction).
More practice: warm-up
1What is the SI unit of magnetic field?
2When does a current-carrying wire feel no force in a magnetic field?
When it is parallel to the field.
3Write the formula for the field inside a solenoid.
4What shape are the field lines around a long straight wire?
Circles centred on the wire.
5Do magnetic field lines start and end anywhere?
No. They form closed loops.
More practice: on your own
1How many turns are needed for a 30 cm solenoid carrying 3.0 A to have a field of 0.010 T inside?
N = BL/(μ₀I) = (0.010)(0.30)/((4π × 10⁻⁷)(3.0)) = 796 turns (about 800).
2At what distance from a wire carrying 10 A is its field equal to Earth's (5.5 × 10⁻⁵ T)?
r = μ₀I/(2πB) = (2 × 10⁻⁷)(10)/(5.5 × 10⁻⁵) = 0.036 m (3.6 cm).
3A high-voltage line 20 m above the ground carries 1000 A. Find the field it makes at ground level and compare with Earth's.
B = (2 × 10⁻⁷)(1000)/20 = 1.0 × 10⁻⁵ T, about one fifth of Earth's field. (The line's current is AC, so this field alternates, while Earth's is steady.)
4Two long parallel wires 4.0 cm apart carry 6.0 A in opposite directions. Find the magnetic field at the point midway between them.
Each wire gives B = (2 × 10⁻⁷)(6.0)/0.020 = 6.0 × 10⁻⁵ T. With opposite currents, the fields at the midpoint point the same way (check with RHR #1), so B = 1.2 × 10⁻⁴ T.
5Compare the field diagrams of a straight conductor and a solenoid, and match each to an electric field arrangement from earlier in the unit.
Straight conductor: circles around the wire, weaker farther away (B ∝ 1/r), a bit like a point charge's field getting weaker with distance. Solenoid: parallel, evenly spaced lines inside (uniform), like the field between parallel plates; outside it resembles a bar magnet. Unlike electric field lines, all magnetic lines are closed loops.
Challenge
1A 30 m horizontal power line carries 500 A from west to east. Earth's field there has a horizontal component of about 1.8 × 10⁻⁵ T pointing north and a vertical component of about 5.2 × 10⁻⁵ T pointing down. Find the size and direction of the force from each component. Does it matter for the line?
Horizontal component (north), current east: RHR #3, thumb east, fingers north, palm up: F = ILB = (500)(30)(1.8 × 10⁻⁵) = 0.27 N upward. Vertical component (down), current east: thumb east, fingers down, palm faces north: F = (500)(30)(5.2 × 10⁻⁵) = 0.78 N north. These forces are tiny compared with the weight of 30 m of cable (hundreds of newtons), so they do not matter.
2A hospital MRI magnet is a superconducting solenoid producing 1.5 T. If it were a simple long solenoid, how many turns per metre would it need for a current of 400 A?
N/L = B/(μ₀I) = 1.5/((4π × 10⁻⁷)(400)) = about 3.0 × 10³ turns per metre. Superconducting wire carries such currents with no resistance, so the magnet does not overheat.
3Explain why two parallel currents in the same direction attract, using RHR #1 and RHR #3.
By RHR #1, wire 1's field at wire 2 is perpendicular to both the wire and the line joining them. Apply RHR #3 to wire 2 (thumb along I₂, fingers along wire 1's field): the palm points toward wire 1. The same argument for wire 1 in wire 2's field gives a force toward wire 2. So they attract. Reverse one current and both forces reverse: they repel.
Quiz
1. The force on a current-carrying wire in a magnetic field is greatest when the wire is
2. A long straight wire's magnetic field 1 cm away is B. At 2 cm it is
3. One tesla is equivalent to
4. Two parallel wires carry currents in opposite directions. They
5. Doubling the number of turns of a solenoid of the same length, with the same current, makes the field inside
6. How are magnetic field lines different from electric field lines?
Big idea (D2.4, D3.3): currents make magnetic fields (B = μ₀I/2πr for a wire, B = μ₀NI/L in a solenoid), and magnetic fields push on currents (F = ILB sin θ, direction by the right-hand rule).
How to help: Ask your teen to show you the right-hand rules with their actual hand while explaining an example. If you have a phone with a magnetometer app, the measurement activity is safe for a few seconds at a time with a single cell; supervise it.
Common difficulties:
- Using r² for a straight wire (it is 1/r).
- Using the left hand by mistake, or mixing electron flow and conventional current.
- Forgetting sin θ, or using the angle between the wire and the force instead of the wire and the field.
- Thinking parallel same-direction currents repel (they attract).
Safety: short-circuiting a cell makes the wire warm; keep contact to a few seconds, use only one AA or D cell, and never use wall outlets or car batteries.