Q.A horizontal straight wire $10\ \text{m}$ long extending from east to west is falling with a speed of $5.0\ \text{m s}^{-1}$, at right angles to the horizontal component of the earth's magnetic field, $0.30 \times 10^{-4}\ \text{Wb m}^{-2}$.

Concept: Motional EMF — when a conductor moves perpendicular to a magnetic field, an emf is induced across its ends given by $\mathcal{E} = B l v$.

Reasoning

1. The wire moves at right angles to the horizontal component of Earth’s field, so the motional emf formula applies directly: $\mathcal{E} = B l v$.
2. Substitute values: $B = 0.30 \times 10^{-4}\ \text{Wb m}^{-2}$, $l = 10\ \text{m}$, $v = 5.0\ \text{m s}^{-1}$.
3. $\mathcal{E} = (0.30 \times 10^{-4}) \times 10 \times 5.0 = 1.5 \times 10^{-3}\ \text{V} = 1.5\ \text{mV}$. …

The falling wire cuts the horizontal component of Earth's magnetic field, inducing a motional emf $\mathcal{E}=B_H\,l\,v$ — valid because the wire, its velocity, and the field are mutually perpendicular.

1. Magnitude of the induced emf With $B_H = 0.30\times10^{-4}\ \text{T}=3.0\times10^{-5}\ \text{T}$, $l=10\ \text{m}$, $v=5.0\ \text{m s}^{-1}$:

   $\mathcal{E}=B_H\,l\,v=(3.0\times10^{-5})(10)(5.0)=1.5\times10^{-3}\ \text{V}=1.5\ \text{mV}.$

2. Direction of the emf Take east $=\hat{i}$, north $=\hat{j}$, up $=\hat{k}$. The velocity is $\vec v=-v\hat k$ (downward) and the horizontal field points north, $\vec B=B_H\hat j$. The force per unit charge on a positive carrier is …

Method: Motional EMF Formula (for a straight conductor moving in a uniform magnetic field)

This method uses the fact that when a conductor cuts magnetic field lines, an emf is induced across its ends.

Steps

Step 1: Identify the given data

• Length of wire, $l = 10\ \text{m}$
• Speed of fall, $v = 5.0\ \text{m s}^{-1}$
• Horizontal component of Earth's magnetic field, $B_H = 0.30 \times 10^{-4}\ \text{Wb m}^{-2}$
• The wire moves perpendicular to the magnetic field → $\theta = 90^\circ$

Step 2: Write the motional emf formula

The induced emf in a straight conductor moving in a uniform magnetic field is:

where $\theta$ is the angle between the velocity vector and the magnetic field.

Step 3: Substitute values

Since $\sin 90^\circ = 1$:

Step 4: Answer (a)

Step 5: Determine direction of induced emf (b)

Use Fleming's Right-Hand Rule (generator rule):

• Thumb: direction of motion (downward, since wire is falling)
• Index finger: direction of magnetic field (horizontal, from south to north — Earth's horizontal component points geographic north)
• Middle finger: direction of induced current (and hence emf) …

Here are the most common mistakes students make with this classic Motional EMF problem, and how to avoid each one.

Mistake 1: Using the Wrong Formula or Forgetting the Perpendicular Condition

The Mistake:

Students often plug numbers into $\varepsilon = Blv$ without checking if the velocity is perpendicular to both the wire and the magnetic field. Some use $\varepsilon = Blv \sin\theta$ but get the angle wrong.

Why it happens:

The formula $\varepsilon = Blv$ is a special case of $\varepsilon = Blv \sin\theta$, valid only when $v$, $B$, and the wire are mutually perpendicular. In this problem, the wire is horizontal (east–west), the velocity is vertical (downward), and the magnetic field is horizontal (northward). These three are indeed mutually perpendicular — so $\sin 90^\circ = 1$.

How to avoid:

• Always draw a 3D sketch:
  • Wire along East–West
  • Velocity downward
  • $B$ (horizontal component) Northward
• Confirm that each pair is perpendicular. If any angle is not $90^\circ$, use $\varepsilon = Blv \sin\theta$ with the correct angle between $v$ and $B$.

Correct calculation:

Mistake 2: Confusing the Direction of Induced EMF (Lenz’s Law vs. Right-Hand Rule)

The Mistake:

Students apply Fleming’s Right-Hand Rule incorrectly — often pointing the thumb in the direction of motion but forgetting that the rule applies to a conductor moving in a magnetic field, not to a current-carrying wire.

Why it happens:

There are multiple right-hand rules (for generators, for motors, for magnetic fields around wires). Mixing them up gives the wrong direction.

How to avoid:

• Use Fleming’s Right-Hand Rule specifically for generators:
  • Thumb = direction of motion (downward)
  • Index finger = magnetic field (northward)
  • Middle finger = induced current (comes out perpendicular to both)
• For this setup: thumb down, index north → middle points west.
• So the induced current flows from east to west inside the wire.

Direction of EMF:

The induced EMF drives current from east to west, so the EMF direction is from east to west along the wire.

Mistake 3: Getting the Higher Potential End Wrong

The Mistake:

Students think the end where current “comes out” is at higher potential, or they confuse the direction of conventional current with electron flow.

Why it happens:

Inside a source of EMF (like a battery or this moving wire), conventional current flows from lower to higher potential — opposite to what happens in a resistor. This is a common conceptual trap.

How to avoid:

• Remember: Inside a source, current flows from negative to positive (low to high potential).
• Here, current flows from east to west inside the wire.
  • So the west end is where current exits the source → higher potential.
  • The east end is where current enters → lower potential. …