Force on a current-carrying conductor, force between parallel currents and the ampere

Force on a current-carrying conductor

A current in a wire is a stream of moving charges. Each moving charge in a magnetic field feels a force, so the whole wire feels a force. For a straight wire of length l carrying current I in a uniform field B, F = I(l × B). Here l is a vector of length l pointing along the current. Its size is F = IlB sin θ, where θ is the angle between the wire and B. The force is zero when the wire is parallel to B and largest (IlB) when it is perpendicular to B. Fleming's left-hand rule gives the direction. Hold the thumb, first finger and middle finger of your left hand at right angles to one another. The first finger points along the field, the middle finger along the current, and the thumb gives the force (motion).

Derivation: F = I(l × B)

A straight rod of length l and cross-sectional area A is in a uniform field B. It has n mobile charge carriers per unit volume, each of charge q, moving with drift velocity v_d.

1. 1
   Number of charge carriers in the rod = nAl.
2. 2
   Force on one carrier = q(v_d × B).
3. 3
   Total force F = (nAl) q(v_d × B).
4. 4
   Current I = nqAv_d, and the current points along v_d for positive carriers. So nqAl v_d = Il, where l points along the current.
5. 5★
   So, F = I(l × B), and its size is F = IlB sin θ. The same result holds for electrons, whose charge and drift velocity are both reversed.

★ key step

Derivation: force between two parallel currents

Two long straight parallel wires, 1 and 2, are a distance d apart in vacuum. They carry currents I₁ and I₂ in the same direction.

1. 1
   Wire 1 makes a field at wire 2: B₁ = μ₀I₁/2πd, perpendicular to wire 2 (right-hand thumb rule).
2. 2
   Wire 2 carries current I₂ in this field. A length L of it feels a force F₂₁ = I₂LB₁, since sin 90° = 1.
3. 3
   So F₂₁ = μ₀I₁I₂L/2πd.
4. 4
   Fleming's left-hand rule shows that this force on wire 2 points towards wire 1. By Newton's third law, wire 1 is pulled towards wire 2 with an equal force.
5. 5★
   Force per unit length: f = μ₀I₁I₂/2πd.
6. 6★
   So, parallel currents attract, and antiparallel (opposite) currents repel.

★ key step

Definition of the ampere

Put I₁ = I₂ = 1 A and d = 1 m in f = μ₀I₁I₂/2πd. Then f = (4π × 10⁻⁷)/(2π) = 2 × 10⁻⁷ N per metre. This gives the textbook definition. Take two very long, straight, parallel conductors of negligible cross-section, placed 1 m apart in vacuum. One ampere is the steady current in each that produces a force of 2 × 10⁻⁷ N per metre of length between them. Note: since 2019 the SI defines the ampere by fixing the value of the electron charge e. For the board exam, write the force definition above.

Worked example: forces on wires

How to draw the parallel-wire diagram

• Draw two long vertical lines for wires 1 and 2, a distance d apart. Put upward arrows on both for currents I₁ and I₂.
• At wire 2 (on the right), show B₁ as ⊗, into the page.
• Draw the force F₂₁ on wire 2 pointing towards wire 1, and F₁₂ on wire 1 pointing towards wire 2.
• Label d, I₁, I₂ and B₁, and write 'attract' between the wires.