doPhysics · Electromagnetism 2
Magnetic Force on Currents and the Magnetic Dipole
The force on a wire carrying a current, straight or bent, and between two parallel currents. The torque that turns a current loop, its magnetic moment and energy, and the moving-coil galvanometer built on that torque.
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Why this chapter
Why this chapter exists
A magnetic field pushes a moving charge sideways. A current is a flood of moving charges, so a wire carrying a current is pushed too, and that push drives every electric motor. This chapter adds up the pushes: on a straight wire, on a wire of any shape, and on one wire by another. Then it closes the wire into a loop. A loop feels no net force in a uniform field, but it turns, like a compass needle. Its turning, measured against a spring, is how a galvanometer reads a current.
You will learn
By the end of this chapter you will
- Build the force on a wire from the drifting electrons, \(d\vec F = I\,d\vec l \times \vec B\), and use \(F = BIL\sin\theta\) with its direction.
- Find the force on a wire of any shape in a uniform field from its effective length. Explain why a closed loop feels no net force.
- Find the force between two parallel currents, state the classic definition of the ampere, and find the force on a loop beside a long wire.
- Find the magnetic moment \(M = NIA\) of a coil and the torque \(\vec\tau = \vec M \times \vec B\) on it, and tell stable from unstable equilibrium.
- Use the energy \(U = -\vec M \cdot \vec B\), the period of small swings \(2\pi\sqrt{I/MB}\), and the moment \(evr/2\) of an orbiting electron.
- Explain the radial field and the soft iron core of a moving-coil galvanometer, and use \(NIAB = k\theta\), its sensitivities and its figure of merit.
Before you begin
Before you begin
From the chapter Magnetic Force on Moving Charges, you should know the force \(q\vec v \times \vec B\) and the right-hand rule for a cross product. You should know current as \(I = neAv_d\) from the chapter Electric Current and Resistance. From Mechanics you need torque as \(\vec r \times \vec F\), the couple, stable and unstable equilibrium, and angular SHM with \(T = 2\pi\sqrt{I/\kappa}\).
Lessons
1
From Moving Charges to a Current
A wire carrying a current is a crowd of drifting electrons, and each one feels the magnetic force. Added up, they give \(dF = I\,dl \times B\), and for a straight wire \(F = BIL\sin\theta\), pushed at right angles to both the wire and the field.
Force on a current elementForce on a straight wireDirection of the force on a wire
2
Wires of Any Shape
In a uniform field a bent wire feels the same force as a straight wire joining its ends, so a closed loop feels no net force. An arc can be integrated directly, and a field that changes from place to place breaks the shortcut.
The effective length ruleNo net force on a closed loopIntegrating along an arcA loop in a non uniform field
3
Parallel Wires and the Ampere
Each of two parallel currents sits in the field of the other, so each is pushed. Like currents attract and unlike currents repel, with \(\mu_0 I_1 I_2/2\pi d\) on every metre. This force once defined the ampere, and it pulls a loop towards a long wire.
Force between parallel currentsThe definition of the ampereA loop beside a long wire
4
Torque on a Loop: the Magnetic Dipole
A current loop has a magnetic moment \(M = NIA\) along its normal. In a uniform field the forces on its sides form a couple, so the loop feels a torque \(\tau = M \times B\) that turns \(M\) towards \(B\). The torque vanishes at 0° and 180°, but only 0° is stable.
Magnetic moment of a loopTorque on a current loopEquilibrium of a dipole
5
Energy of a Dipole and the Circulating Electron
Turning a dipole against its torque stores energy, \(U = -M \cdot B\), lowest with \(M\) along \(B\). Near that position the dipole swings like a torsion pendulum, with \(T = 2\pi\sqrt{I/MB}\). An electron in orbit is itself a tiny current loop, with \(M = evr/2\).
Potential energy of a dipoleSmall oscillations of a dipoleThe orbiting electron as a dipole
6
The Moving-Coil Galvanometer
A coil hung in a radial field between curved pole pieces turns until the magnetic torque \(NIAB\) balances the spring's \(k\theta\). So the deflection is proportional to the current, and the scale is even. Sensitivity and the figure of merit follow from the same balance.
The radial field and the soft iron coreBalance of torques in the galvanometerSensitivity and figure of meritWhy the scale is linear
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- ampere
- The SI unit of current. It was once defined by the force between two long parallel wires \(1\,\text{m}\) apart, \(2 \times 10^{-7}\,\text{N}\) on each metre. Since 2019 it is fixed through the charge \(e\).
- couple
- A pair of equal and opposite forces acting along different lines. It has no net force but a torque equal to one force times the distance between the lines.
- current element
- A short length \(d\vec l\) of a wire carrying a current \(I\), written \(I\,d\vec l\) and pointing along the current. In a field it feels \(d\vec F = I\,d\vec l \times \vec B\).
- current sensitivity
- The deflection of a galvanometer per unit current, \(\theta/I = NAB/k\).
- effective length
- The straight vector \(\vec L_{eff}\) from the start of a wire to its end. In a uniform field a wire of any shape feels \(I\vec L_{eff} \times \vec B\).
- figure of merit
- The current that gives a deflection of one division on a galvanometer's scale, such as \(0.1\,\text{mA}\) per division. A smaller figure of merit means a more sensitive galvanometer.
- Fleming's left-hand rule
- A rule for the direction of the force on a current in a field. Hold the thumb and first two fingers of the left hand at right angles. With the first finger along the field and the second along the current, the thumb gives the force.
- gyromagnetic ratio
- The ratio of the magnetic moment of a spinning or orbiting charged body to its angular momentum. For an orbiting electron it is \(e/2m_e\).
- magnetic dipole
- A current loop, a magnet or any body that feels no net force in a uniform magnetic field. It feels only a torque that turns it to line up with the field.
- magnetic moment
- The vector \(\vec M = NI\vec A\) of a coil, along its normal by the right-hand rule, in A m². It fixes the torque \(\vec M \times \vec B\) and the energy \(-\vec M \cdot \vec B\) of the dipole in a field.
- permeability of free space
- The constant \(\mu_0 = 4\pi \times 10^{-7}\,\text{T m/A}\) that sets the strength of the magnetic field made by a current in vacuum. A long wire has \(B = \mu_0 I/2\pi d\).
- phosphor bronze
- An alloy of copper, tin and a little phosphorus, used for the spring or suspension strip of a galvanometer. It has a small torsion constant, is very elastic and non-magnetic, and conducts.
- radial field
- A magnetic field whose lines run along radii from a central axis. It fills the gap between the curved pole pieces and the iron core of a galvanometer. A coil turning in it always has its plane along the field.
- torsion constant
- The restoring torque per radian of twist of a spring or suspension strip, \(k = \tau/\theta\), in N m/rad.
- voltage sensitivity
- The deflection of a galvanometer per unit potential difference across it, \(\theta/V = NAB/kR\), where \(R\) is the resistance of its coil.