Magnetic Fields from Current-Carrying Conductors
Hans Christian Oersted (Copenhagen, April 1820): during a lecture, Oersted connected a wire to a battery and noticed a compass needle beneath the wire deflected. With 10 A flowing through the wire, a compass 1 cm away experienced a field of about 200 μT — four times Earth's background field. This accidental discovery united electricity and magnetism for the first time: $B = \mu_0 I / (2\pi r)$.
A long straight wire carries a current of 5.0 A. A compass is placed 2 cm from the wire.
Predict 1: In what shape do the magnetic field lines form around the wire?
Predict 2: If the current is doubled to 10 A, how does the field strength at 2 cm change?
The magnetic field lines around a long straight current-carrying conductor form:
Know
- Field around a long straight wire: $B = \mu_0 I / (2\pi r)$
- $\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}$ (permeability of free space)
- Right-hand grip rule for direction of field around a wire
- Field forms concentric circles; strength decreases with distance $r$
Understand
- Why $B \propto I$ and $B \propto 1/r$
- How the right-hand grip rule works for both straight wires and solenoids
- Historical significance: Oersted's discovery unified electricity and magnetism
Can Do
- Calculate $B$ at a distance $r$ from a long straight wire
- Determine direction of $\boldsymbol{B}$ using right-hand grip rule
- Sketch the field pattern around a straight wire (dots/crosses for current direction)
Core Content
Lay a compass on a table. It points north. Now run a wire directly above the compass and connect it to a battery so that current flows — the compass needle swings sideways, deflecting away from north. Switch off the current and it swings back. Reverse the current and it deflects the opposite way. The wire is doing something the table, the battery, and the air cannot: it is creating a magnetic field in the space around it. This is what Hans Christian Oersted observed in Copenhagen in April 1820 — and it is the reason every electric motor, generator, and MRI machine exists.
$B = \dfrac{\mu_0 I}{2\pi r}$
$B$: field strength (T); $\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}$; $I$: current (A); $r$: perpendicular distance from wire (m)
$B \propto I$ — doubling current doubles the field strength.
$B \propto 1/r$ — doubling distance halves the field strength.
Calculate $B$ at $r = 0.01\ \text{m}$ from a wire carrying $I = 10\ \text{A}$. (Oersted's 1820 setup.)
- $B = \dfrac{\mu_0 I}{2\pi r} = \dfrac{4\pi \times 10^{-7} \times 10}{2\pi \times 0.01}$
- $= \dfrac{4 \times 10^{-6}}{0.02\pi} = \dfrac{4\pi \times 10^{-6}}{2\pi \times 0.01}$
- $= \dfrac{4\pi \times 10^{-6}}{2\pi \times 10^{-2}} = \dfrac{4 \times 10^{-6}}{2 \times 10^{-2}} = 2.0 \times 10^{-4}\ \text{T} = 200\ \mu\text{T}$
- Earth's field $\approx 50\ \mu\text{T}$ → this wire produces 4× Earth's background field ✓
Right-hand grip rule
Point thumb of right hand in direction of conventional current (+ve to −ve). Fingers curl in the direction of $\boldsymbol{B}$ (concentric circles around the wire).
A long straight wire carrying current $I$ produces a magnetic field $B = \mu_0 I / (2\pi r)$ (T), where $\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}$. Field lines form concentric circles; $B \propto I$ and $B \propto 1/r$. Direction: right-hand grip rule — thumb along current, fingers curl in field direction.
Pause — copy the highlighted formula and rule into your book before the check below.
Doubling the current in a wire doubles the magnetic field strength at a fixed distance from the wire.
The magnetic field lines around a straight wire are straight lines parallel to the wire.
A long straight wire carries a current of 20 A.
- Calculate the magnetic field strength at a distance of 5.0 cm from the wire.
- Calculate the magnetic field strength at 10 cm from the wire.
- Show that the ratio of the two field strengths equals the inverse ratio of the distances.
At a distance of 4.0 cm from a long straight wire carrying 8.0 A, the magnetic field strength is approximately:
A vertical wire carries conventional current upward (out of the floor, into the ceiling).
- Using the right-hand grip rule, determine the direction of the magnetic field at a point to the east of the wire (at the same height).
- If a compass is placed to the east of the wire, in which direction does the compass needle point?
- Explain why Oersted's observation was significant — what previous assumption did it challenge?
Three of these are correct statements about the field around a straight wire. Pick the odd one out.
The magnetic field at 2.0 cm from a wire carrying 5.0 A is _____ × 10⁻⁵ T. (Use $\mu_0 = 4\pi \times 10^{-7}$ and round to 2 sig figs; enter the coefficient only.)
A current-carrying wire is oriented so current flows into the page. The magnetic field at a point directly above the wire points:
ApplyBand 3(3 marks) 3. A long straight wire carries a current of 15 A. Calculate the magnetic field strength (a) at 1.0 cm, (b) at 5.0 cm, and (c) at 10 cm from the wire. Express your answers in μT.
AnalyseBand 4(3 marks) 4. Two long parallel wires, 6.0 cm apart, carry currents of 10 A (wire A) and 20 A (wire B) in the same direction. (a) Calculate $B_A$ and $B_B$ at the midpoint between the wires. (b) Determine the direction of each field at the midpoint using the right-hand grip rule. (c) Calculate the net field at the midpoint.
EvaluateBand 6(4 marks) 5. Oersted's discovery that a current-carrying wire deflects a compass was a pivotal moment in physics. Using the relationship $B = \mu_0 I / (2\pi r)$ and your knowledge of scientific models, evaluate the significance of Oersted's observation and explain what it revealed about the relationship between electricity and magnetism.
Show all answers
Activity 1 — Model Answers
- $B(5\ \text{cm}) = \dfrac{4\pi \times 10^{-7} \times 20}{2\pi \times 0.05} = \dfrac{8 \times 10^{-6}}{0.1\pi} = \dfrac{8 \times 10^{-6}}{0.314} \approx 8.0 \times 10^{-5}\ \text{T} = 80\ \mu\text{T}$
- $B(10\ \text{cm}) = \dfrac{4\pi \times 10^{-7} \times 20}{2\pi \times 0.10} = 4.0 \times 10^{-5}\ \text{T} = 40\ \mu\text{T}$
- $B_1/B_2 = 80/40 = 2$; $r_2/r_1 = 10/5 = 2$ ✓ — confirms $B \propto 1/r$
Activity 2 — Model Answers
- Right-hand grip rule: thumb points upward (current direction). At a point to the east, the fingers point south (into the south side of the wire). The field direction is south.
- The compass needle aligns with the field and points south (its north end toward south).
- Before Oersted, it was believed electricity and magnetism were completely independent phenomena. Oersted showed that an electric current (moving charges) produces a magnetic field — establishing the first experimental link between the two. This overturned the prevailing view and paved the way for electromagnetism as a unified theory.
Short Answer — Model Answers
Q3 (3 marks): Using $B = \mu_0 I / (2\pi r)$ with $I = 15\ \text{A}$: (a) $r = 0.010\ \text{m}$: $B = (4\pi \times 10^{-7} \times 15)/(2\pi \times 0.01) = 3.0 \times 10^{-4}\ \text{T} = 300\ \mu\text{T}$. (b) $r = 0.050\ \text{m}$: $B = 6.0 \times 10^{-5}\ \text{T} = 60\ \mu\text{T}$. (c) $r = 0.10\ \text{m}$: $B = 3.0 \times 10^{-5}\ \text{T} = 30\ \mu\text{T}$.
Q4 (3 marks): Midpoint is 3.0 cm from each wire. (a) $B_A = \mu_0 \times 10/(2\pi \times 0.03) = 6.67 \times 10^{-5}\ \text{T} \approx 67\ \mu\text{T}$; $B_B = \mu_0 \times 20/(2\pi \times 0.03) = 1.33 \times 10^{-4}\ \text{T} \approx 133\ \mu\text{T}$. (b) For wires carrying current in the same direction: at the midpoint, $B_A$ (from wire A) points in one direction and $B_B$ (from wire B) points in the opposite direction — they partially cancel. (c) Net $B = 133 - 67 = 66\ \mu\text{T}$ (direction of $B_B$).
Q5 (4 marks): Oersted's discovery was significant for several reasons. Quantitatively, $B = \mu_0 I/(2\pi r)$ shows that $B \propto I$ — any electric current, no matter how small, produces a measurable magnetic field in its surroundings. This was entirely unexpected in 1820, when electricity (charge in motion) and magnetism (lodestones, compass needles) were classified as separate forces. Oersted's observation — a compass deflecting near a current-carrying wire — demonstrated for the first time that moving electric charge produces a magnetic field, suggesting a fundamental connection between the two. This launched a new era: Ampere quantified the relationship (Ampere's law), Faraday discovered electromagnetic induction (the reverse: changing magnetic fields produce currents), and Maxwell unified all of electromagnetism in four equations. Without Oersted's accidental discovery, the electric motor, generator, and transformer would not have been developed.
Five timed questions on magnetic fields from current-carrying conductors.
⚔ Enter the arenaHans Christian Oersted, Copenhagen, April 1820: a wire carrying 10 A produced B = μ₀ × 10/(2π × 0.01) = 200 μT at 1 cm — four times Earth's own 50 μT field. His compass deflected 90°. A single lecture demonstration ended the belief that electricity and magnetism were unrelated forces, and within months Ampere and Biot-Savart had derived the formula B = μ₀I/(2πr) that quantifies it.
Now check your Think First answers: field lines form concentric circles. Doubling current doubles field strength (B ∝ I) — confirmed by the formula. At 2 cm from a 5 A wire: B = μ₀ × 5/(2π × 0.02) = 50 μT.