Magnetic effect of current

Oersted's experiment, qualitatively. A straight wire carries current past a compass. The needle deflects from its normal north-south rest position. Reverse the current, and the needle deflects the opposite way. Increase the current, and the deflection grows. None of this needs a formula — it is a historical, qualitative observation, and the CISCE syllabus explicitly excludes "Ampere's swimming rule" (a mnemonic for predicting the deflection) from the theory content. What matters is the conclusion: a current is a source of magnetic field, exactly as a bar magnet is.

Two questions, one chapter. Once you accept that currents make fields and fields push on currents, the whole chapter is just two questions being answered in turn. First — given a current, what field does it make? The Biot-Savart law gives the field from any small element of current, and integrating it over standard shapes yields ready-made formulas for a straight wire, a circular loop, and a solenoid. For situations with enough symmetry, Ampere's circuital law gets there faster:

where: $\vec B$ is the magnetic field, $d\vec l$ a small element along a closed loop (an "Amperian loop"), and $I_{\text{enc}}$ the current actually threading that loop; $\mu_0 = 4\pi\times10^{-7}\ \text{T m/A}$ is the permeability of free space, the constant that fixes how strongly current sources a field in vacuum. Second — given a field, what force does it exert? The Lorentz force answers it for a single moving charge, and the force on a current-carrying wire follows from summing that force over all the charges drifting inside it:

where: $q$ is the charge, $\vec v$ its velocity, $\vec E$ and $\vec B$ the electric and magnetic fields, $I$ the current in a straight segment, and $\vec l$ a vector along the wire in the direction of current flow, of length equal to the segment. Two current-carrying wires side by side then exert forces on each other through exactly this mechanism, and a current loop placed in a field feels a net torque, $\vec\tau = \vec m\times\vec B$, which is the working principle of the moving-coil galvanometer.

Everything is a cross product. Both halves of the story — field-from-current and force-from-field — involve a vector cross product, so direction is always found with a right-hand-rule construction: one version (curl the fingers, thumb along current) gives field direction; a different version (fingers along $\vec v$, curl toward $\vec B$, thumb gives force) gives force direction. Keeping these two constructions straight from day one saves confusion in every later topic.

There is no single number-crunching derivation at this stage of the chapter — Oersted's experiment is qualitative, and the quantitative laws it motivates (Biot-Savart, Ampere, Lorentz force, torque) each get their own dedicated article. What is worth reasoning through here is why a right-hand-rule habit, rather than memorising separate rules for every shape of wire, is enough to survive the whole chapter.

Step 1. Every magnetic effect in this chapter traces back to one of two cross products: $\vec B$ from a current element ($d\vec B \propto d\vec l\times\hat r$ in the Biot-Savart law) or force from a field ($\vec F = q\vec v\times\vec B$, or its wire version $\vec F = I\vec l\times\vec B$). A cross product's direction is fixed by the right-hand rule, no exceptions.

Step 2. The first cross product is a "curl" construction — point the thumb along the current, and the fingers show how $\vec B$ circles it. This single grip handles a straight wire, a loop, and a solenoid alike; only the geometry you apply it to changes.

Step 3. The second cross product is a "sweep" construction — point the fingers along $\vec v$ (or along the wire, for $\vec F = I\vec l\times\vec B$), curl them toward $\vec B$, and the thumb gives $\vec F$. This is a genuinely different hand motion from Step 2, which is exactly why students conflate them.

Step 4. Because both constructions are the same right-hand rule applied to different vector pairs, mastering the geometry of a cross product once — rather than memorising Oersted's needle-deflection direction, then the solenoid's polarity, then the force on a wire, as unrelated facts — carries you through the entire chapter with one reusable skill.

A long straight vertical wire carries current flowing upward. A compass sits on a horizontal table, due east of the wire. Later, the same wire is laid horizontally, running east-west, directly above a second compass, still carrying current — now from west to east. Find the field direction at each compass.

Step 1. Vertical wire, compass to the east. Grip the wire with the right thumb pointing upward, along the current. Viewed from above, the fingers curl counter-clockwise. At a point due east of the wire, this circulation gives a field tangent to the circle there, pointing north.

Step 2. Horizontal wire, compass directly below. Grip the wire with the thumb pointing east, along the current. The fingers curl so that below the wire the field points north, and above it points south.

Step 3 — check by reversal. In both cases, reversing the current direction flips the field to the opposite compass direction — this reversal test is exactly what Oersted himself used to confirm the effect was real and not coincidence. Answer: the vertical wire's field points north at the compass to its east; the horizontal wire's field points north at the compass directly below it — and reversing either current reverses both deflections.

Competitive stretch

If the same horizontal wire instead carried current from east to west (reversed), the field directly below it would point south, now opposing Earth's own northward field component there rather than reinforcing it — a detail exam-setters like to flip to test whether you re-apply the grip rule each time rather than recalling "the answer was north" from memory.

• Reaching for "Ampere's swimming rule" — it is explicitly excluded from the CISCE syllabus; use the right-hand grip rule for field direction instead.
• Treating Oersted's experiment as needing a calculation — it is a qualitative, historical observation, not a source of numbers.
• Mixing up the two right-hand-rule constructions: grip-the-wire gives field direction; fingers-along-$\vec v$-curl-toward-$\vec B$ gives force direction on a moving charge or wire.
• Forgetting that reversing the current reverses the field (and hence the deflection) — this reversal is the core evidence, and a common exam question asks you to predict it.

Ask the tutor: Not sure which right-hand rule applies to a given question, or want a guided walkthrough connecting Oersted's original observation to the Biot-Savart law, Ampere's law, and the force laws that follow — ask the AI tutor to map the chapter's logic with you.