Q.Using Ampere's circuital law, derive the expression for the magnetic field at a perpendicular distance $r$ from a long, straight, current-carrying conductor, explaining the choice of the circular Amperian loop.

The field of a straight wire at a point is along $\hat{I} \times \hat{r}$. With the current along $-\hat{j}$ and the point on the positive $x$-axis ($\hat{r} = +\hat{i}$), this gives $(-\hat{j}) \times \hat{i} = +\hat{k}$ — the field points along the $z$-axis, option (A).

By the right-hand (thumb) rule, the magnetic field produced by a current flowing from South to North, at a point directly below the wire, points towards the West — so the compass needle (normally pointing North) deflects towards the West, i.e. to the left, when viewed in the conventional map orientation (North at top, East to the right).

Setting up directions

Take the standard compass/map convention: North $=+\hat y$, East $=+\hat x$, Up (vertically, away from the ground) $=+\hat z$. This is a right-handed set: $\hat x\times\hat y=\hat z$.

The wire is aligned along the North–South line and carries current from S to N, i.e. the current direction is $\hat I = +\hat y$ (Northward).

The compass needle sits directly below the wire — i.e. at a point displaced from the wire in the $-\hat z$ direction (vertically downward, towards the ground).

Applying the Biot–Savart / right-hand rule

For an infinite straight current-carrying wire, the magnetic field at a point is directed along $\hat I \times \hat r$, where $\hat r$ is the unit vector from the wire to the field point.

Here $\hat r = -\hat z$ (point is directly below the wire), so:

$\vec B \ \propto\ \hat I\times\hat r = \hat y\times(-\hat z) = -(\hat y\times\hat z) = -\hat x$

Since $+\hat x=$ East, $-\hat x=$ West. So the magnetic field produced by the wire, at the point directly below it, points towards the West.

Effect on the compass needle

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At the midpoint between two parallel wires carrying equal currents in the same direction, each wire's field has equal magnitude but the right-hand rule shows the two fields point in opposite directions there — so they cancel and the net field is zero.

Setup

Two long, straight, parallel wires, separated by $2r$, each carrying the same current $I$ in the same direction. Consider the point exactly midway between them — a distance $r$ from each wire.

Field due to each wire

For an infinite straight wire, the field at perpendicular distance $r$ has magnitude

$B=\frac{\mu_0 I}{2\pi r}$

directed in circles around the wire (by the right-hand rule — point the thumb along the current, fingers curl in the direction of $\vec B$).

So each wire, individually, produces a field of the same magnitude $\dfrac{\mu_0 I}{2\pi r}$ at the midpoint (since both are the same distance $r$ from it and carry the same current $I$).

Direction of each field at the midpoint

Let the two wires be parallel to each other, both carrying current in (say) the $+\hat z$ direction, with wire 1 at $x=0$ and wire 2 at $x=2r$; the midpoint is at $x=r$.

• For wire 1, the midpoint lies in the $+\hat x$ direction from the wire. By the right-hand rule ($\vec B \propto \hat I\times\hat r$), $\hat z\times\hat x=\hat y$, so $\vec B_1$ points in $+\hat y$. …

The magnetic field around a long straight current-carrying wire forms concentric circles in a plane perpendicular to the wire. For a vertical wire with upward current, the field lines are horizontal circles centered on the wire — so the correct option is (C).

(b) (i) – (r), (ii) – (q), (iii) – (p). Straight wire $\to \dfrac{\mu_0 I}{2\pi r}$ (r); circular-loop centre $\to \dfrac{\mu_0 I}{2r}$ (q); solenoid $\to \mu_0 n I$ (p). …

Deriving μ from the force law between two current-carrying wires gives the dimensional formula [M L T⁻² A⁻²].

The force per unit length between two long parallel current-carrying wires (carrying currents I_1, I_2, separated by distance d) is:

F/L = μ I_1 I_2 / (2π d)

So μ = 2π d (F/L) / (I_1 I_2). We find the dimensions of each quantity on the right:

• F (force) has dimensions [M L T⁻²]
• d and L (lengths) have dimensions [L]
• I_1, I_2 (currents) have dimensions [A]

Substituting:

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Hans Christian Oersted discovered the magnetic effect of current.

In 1820 Oersted noticed that a compass needle near a current-carrying wire deflected, showing that an electric current produces a magnetic field. This observation, foundational to the NCERT/CBSE Cl …

At a point equidistant from two parallel current-carrying wires, the individual fields either cancel or add depending on the current directions and O's position relative to the wires.

For two long straight parallel wires each carrying the same current I, each produces a field of magnitude B at point O (equidistant from both wires).

• When the geometry is such that the two individual fields at O point in opposite directions (as happens, for example, when O lies symmetrically between two wires carrying current in the same direction), they cancel: resultant = zero. …

Using the right-hand rule for a current flowing west, the field directly above the wire points north.

Take east = $\hat{x}$, north = $\hat{y}$, up = $\hat{z}$ (a right-handed set, since east × north = up). The current flows east to west, i.e. along $-\hat{x}$.

Grip the wire with the right hand, thumb pointing west (direction of current). The fingers curl to show the field direction. At a point directly above the wire, this curl points towards north.

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The magnetic field lines of an infinite straight current-carrying wire are concentric circles around the wire, in a plane perpendicular to it.

For an infinitely long, straight conductor carrying current $I$, the magnetic field at a perpendicular distance $r$ from the wire has magnitude:

$B = \frac{\mu_0 I}{2\pi r}$

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Two standard rules used to find the direction of the magnetic field produced by a current: the Right-Hand Thumb Rule and Maxwell's Cork-Screw Rule.

1. Right-Hand Thumb Rule (Right-Hand Grip Rule): If a current-carrying conductor is held in the right hand such that the thumb points in the direction of current flow, the direction in which the fingers curl around the conductor gives the direction of the magnetic field lines around it.

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By Ampere's circuital law, the field of an infinite straight current-carrying wire is $B=\mu_0I/(2\pi R)$, which is inversely proportional to the distance $R$.

Working

Applying Ampere's circuital law to a circular Amperian loop of radius $R$ centred on a long straight wire carrying current $I$ (by symmetry, $B$ is uniform in magnitude around the loop and tangential to it): …