Physics › Magnetic fields › Magnetic flux density and the force on a wire
Magnetic flux density and the force on a wire
The magnetic field differs from the other two in two ways: its force acts only on moving charge, and it acts at right angles to both the current and the field. One formula, F = BIl sin θ, covers it, the left-hand rule gives the direction, and a top-pan balance measures the force in the lab.
Pick your board and the few notes written for the other boards quietly fold away, here and in the practice players. Nothing is deleted: every folded piece reopens on a tap.
Builds on The field concept and Current, charge and the direction problem.
IN THIS TOPIC
- Use F = BIl for a wire perpendicular to the field.
- Apply Fleming's left hand rule to find the direction of the force.
- Define magnetic flux density and the tesla.
- Describe the required-practical measurement of the force with a top-pan balance.
COMMON MISCONCEPTION
Fields always push along their field lines.
A sideways force
Gravitational fields pull masses along the field lines; electric fields push charges along them. The magnetic field breaks that pattern twice. It acts only on moving charge, meaning a current, and its force points at right angles to both the current and the field. For a straight wire of length l carrying current I across a field of flux density B, the magnitude is
valid when field and current are perpendicular. Direction comes from Fleming's left hand rule. First finger along the Field, seCond finger along the Current, thuMb for the Motion, which is the force, all three held at right angles to one another. Diagrams handle the third dimension with the standard symbols, a circled cross for a field into the page and a circled dot for one coming out.
The right angle is not decoration. A wire lying along the field feels no force whatsoever, and at angles in between only the component of the field perpendicular to the wire does anything. This field never pushes along its own lines.
AQA stops there, examining the perpendicular case alone and printing F = BIl bare in its booklet. Three boards ask for the in-between case as an equation, with θ the angle between the wire and the field,
OCR and Edexcel both print it; CIE does not, so only there is it a recall item. It says nothing new. B sin θ is simply the component of the field across the wire, so θ = 90° gives sin θ = 1 and hands back F = BIl, while θ = 0 gives zero for a wire lying along the field.
Flux density, and the tesla
Rearranging the force law defines the field's strength. The magnetic flux density B is the force per unit current per unit length of perpendicular wire, B = F/Il, and its unit, the tesla, drops straight out of that. One tesla is the flux density that puts one newton on each metre of wire carrying one amp at right angles to the field. A tesla is a substantial field. Laboratory magnets sit in the tens of milliteslas, and the Earth's field is a few hundredths of a millitesla.
WORKED EXAMPLE
Measuring B with a wire
A wire carrying 2.5 A has 8.0 cm inside a magnetic field, at right angles to it, and feels a force of 0.012 N. Find the flux density.
Rearrange the defining equation to B = F/Il.
B = 0.012/(2.5 × 0.080) = 0.060 T.
Sixty milliteslas is a healthy laboratory magnet, and the calculation is the tesla's own definition run backwards, one newton per amp per metre of crossing wire.
The patterns currents make
The force law needs a field to exist in the first place, and currents make their own. Two patterns cover nearly everything you will meet, the field of a straight wire and the field of a wire wound into a solenoid.
Two parallel wires make each other move, and the field patterns say how. Each wire sits in the circular field of the other, so each feels F = BIl. Currents flowing the same way attract and opposite ways repel, as one left-hand-rule check on either wire confirms. CIE names this effect, and it once defined the ampere. Both patterns are read with the right-hand grip rule.
Grip a straight wire with the thumb along the conventional current and the fingers curl the way the field circles it, densest close to the wire and weakening with distance. Curl the fingers the way the current circulates around a solenoid instead, and the thumb points along the core field, out of the north end. An iron core strengthens the field enormously, which is the working principle of the electromagnet.
CIE asks for a third pattern to be sketched, the flat circular coil, and it sits between the other two. Take a single loop lying flat in the plane of the page. Every short piece of that loop circles its own field by the grip rule, and inside the hole all those contributions point the same way, so the field runs through the centre of the coil, straightest and strongest there, then bulges out and loops back round the outside. Draw it as a bar magnet's field squashed flat, one face of the coil north and the other south, which is what a solenoid one turn long amounts to. Curl the fingers the way the current goes round and the thumb points out of the north face, the same grip rule a third time.
Required practical 10, weighing a force
The tenth required practical measures the force with nothing more exotic than a top-pan balance. A magnet sits on the balance and a stiff wire is clamped, separately, between its poles. Switch the current on and the field pushes the wire, say upward; by Newton's third law the wire pushes the magnet downward with an exactly equal force, and the balance reading rises by F/g. Varying the current, the wire length in the field, and the flux density then tests each proportionality in F = BIl in turn, converting grams of reading change back to newtons each time.
GUIDED PRACTICE
Predict the balance
In the RP10 arrangement, the field is 0.060 T, the wire carries 4.0 A, and 5.0 cm sits in the field. Predict the change in the balance reading, in grams.
Show the working
F = BIl = 0.060 × 4.0 × 0.050 = 0.012 N, pressing on the magnets by Newton's third law.
The balance reads mass, so Δm = F/g = 0.012/9.81 = 1.2 × 10−3 kg, a change of 1.2 g and comfortably readable on a 0.01 g balance. Predicting the reading before you switch on is the best habit this practical teaches.
ASSESSMENT FOCUS
- F = BIl holds for the perpendicular arrangement, so say so when you quote it, and give zero force with no calculation at all for a wire lying along the field.
- CIE, Edexcel and OCR all set the general F = BIl sin θ, with θ measured between the wire and the field. OCR and Edexcel print it. CIE makes you recall it, and Pearson prints every formula 9PH0 needs, so an Edexcel candidate memorises none of this. AQA candidates need only the perpendicular form, which is on the booklet, so do not spend memory on the angle if AQA is your board.
- Fleming's left hand needs conventional current. Electron-flow questions are written to catch you, so reverse the direction before your fingers move.
- Define the tesla operationally, as one newton per amp per metre of perpendicular wire. The definition is the rearranged force equation put into words.
- Into-page and out-of-page symbols are examinable vocabulary. Circled cross in, circled dot out. Read them before you start reasoning.
- CIE asks for three field patterns from memory, so practise all three: concentric circles round a long straight wire, a field through the hole of a flat circular coil, and the bar-magnet field of a long solenoid. One right-hand grip rule reads every one of them.
- In the balance practical the mark scheme wants Newton's third law named. The force on the magnet is the equal and opposite partner of the force on the wire, and the reading changes by F/g.
CHECK YOURSELF
A wire carries 3.0 A at right angles across a field of 0.20 T, with 5.0 cm of the wire in the field. Find the force, and the change in reading of a balance supporting the magnet, in grams.
Show a hint
F = BIl in base units, then grams via g.
Show the answer
= 0.20 × 3.0 × 0.050 = 0.030 N.
The magnet feels the equal and opposite partner force, so the reading changes by F/g = 0.030 / 9.81 = 3.1 × 10−3 kg, about 3.1 g.
Whether the reading rises or falls depends on the current's direction; reversing it flips the change, which is itself a check the effect is real.
The magnetic force acts on current, at right angles to everything.
One tesla puts one newton on every metre of crossing wire, per amp.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their mark schemes on the magnetic flux density and the force on a wire questions page.
WHERE TO GO NEXT
- Required practical 10: force on a current-carrying wire puts this topic in the lab, and the written papers ask about it.
- Significant figures, units and checking is the maths this lesson leans on, worked through from GCSE.
CHECK YOUR PROGRESS
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- Use F = BIl for a wire perpendicular to the field.
- Apply Fleming's left hand rule to find the direction of the force.
- Define magnetic flux density and the tesla.
- Describe the required-practical measurement of the force with a top-pan balance.
Open the full revision checklist to track your progress across the whole unit.