State and use the relative directions of the magnetic field, the induced current and the force on a conductor.
Key Concepts
When a conductor moves in a magnetic field, an emf is induced (Faraday’s law).
The magnitude of the induced emf is given by
\$\mathcal{E}=B\,l\,v\sin\theta\$
where \$B\$ is the magnetic flux density, \$l\$ the length of the conductor within the field, \$v\$ the speed of motion and \$\theta\$ the angle between \$v\$ and \$B\$.
The direction of the induced current is not arbitrary – it follows specific right‑hand rules and Lenz’s law.
Stretch the thumb, fore‑finger and middle finger of the right hand so that they are mutually perpendicular.
Thumb – direction of motion of the conductor (\$\vec{v}\$).
Fore‑finger – direction of the magnetic field (\$\vec{B}\$) (from North to South).
Middle finger – direction of the induced current (\$I\$) in the conductor.
2. Lenz’s Law
The induced current always flows in a direction that creates a magnetic field opposing the change that produced it.
3. Fleming’s Left‑Hand Rule (force on a current‑carrying conductor)
Used when the current is already known and we need the direction of the force.
Fore‑finger – direction of magnetic field (\$\vec{B}\$).
Middle finger – direction of current (\$I\$).
Thumb – direction of the force (\$\vec{F}\$) on the conductor.
Using the Rules – Step‑by‑Step Procedure
Identify the direction of the magnetic field lines (\$\vec{B}\$). Usually given as “into the page” (\$\otimes\$) or “out of the page” (\$\odot\$).
Determine the direction of motion of the conductor (\$\vec{v}\$). Use the arrow shown on the diagram.
Apply Fleming’s right‑hand rule to find the direction of the induced current (\$I\$).
Check the result with Lenz’s law: the magnetic field produced by \$I\$ must oppose the original change (e.g., if the flux is increasing into the page, the induced field must be out of the page).
If the problem asks for the force on the moving conductor, use Fleming’s left‑hand rule with the known \$I\$ and \$\vec{B}\$.
Typical Scenarios
Scenario
Magnetic Field \$\vec{B}\$
Motion \$\vec{v}\$
Induced Current \$I\$ (right‑hand rule)
Resulting Force \$\vec{F}\$ (left‑hand rule)
Conductor pulled to the right across uniform field into the page
Into the page (\$\otimes\$)
Right
Upwards (out of the page)
Downwards (opposes motion)
Loop rotating clockwise in a field out of the page
Out of the page (\$\odot\$)
Clockwise rotation (tangential motion)
Current flows anticlockwise when viewed from above
Torque opposes the rotation (Lenz’s law)
Bar moving downwards into a field directed to the right
To the right
Downwards
Current flows from left to right
Force to the left (opposes motion)
Worked Example
Problem: A straight conductor of length \$0.20\,\$m moves at \$5.0\,\$m s\(^{-1}\) to the right through a uniform magnetic field of \$0.30\,\$T directed into the page. Determine the direction of the induced current and the magnetic force on the conductor.
Identify \$\vec{B}\$: into the page (\$\otimes\$).
Identify \$\vec{v}\$: to the right.
Apply Fleming’s right‑hand rule:
Thumb → right, fore‑finger → into page, middle finger → upward (out of the page).
Hence the induced current flows upward (out of the page).
Find the force using Fleming’s left‑hand rule (current known, \$\vec{B}\$ known):
Fore‑finger → into page, middle finger → upward, thumb → to the left.
The magnetic force on the conductor is to the left, opposing the motion, in agreement with Lenz’s law.
Summary Table
Quantity
Symbol / Direction
Rule Used
Key Point
Magnetic field
\$\vec{B}\$ (into/out of page)
Given or drawn
Direction of field lines.
Conductor motion
\$\vec{v}\$
Identify from diagram
Relative to \$\vec{B}\$ determines emf.
Induced current
\$I\$
Fleming’s Right‑Hand Rule
Thumb = \$\vec{v}\$, fore‑finger = \$\vec{B}\$, middle = \$I\$.
Magnetic force on conductor
\$\vec{F}\$
Fleming’s Left‑Hand Rule
Fore‑finger = \$\vec{B}\$, middle = \$I\$, thumb = \$\vec{F}\$.
Direction of induced field
Opposes change in flux
Lenz’s Law
Ensures energy conservation.
Suggested diagram: Conductor moving rightward through a uniform magnetic field into the page, showing \$\vec{v}\$, \$\vec{B}\$, induced current direction, and magnetic force.
Common Mistakes to Avoid
Confusing Fleming’s right‑hand rule (induced current) with the left‑hand rule (force on a current‑carrying conductor).
Neglecting the \$\sin\theta\$ factor when the motion is not perpendicular to the field.
Forgetting Lenz’s law when checking the direction of the induced current – the induced magnetic field must oppose the change.
Assuming the induced emf always drives current in the same direction regardless of the motion; the direction reverses if the motion or field direction is reversed.
Practice Questions
A rectangular loop of wire is pulled out of a magnetic field that points into the page. Indicate the direction of the induced current and the magnetic field it creates.
A rod of length \$0.15\,\$m moves upward at \$3.0\,\$m s\(^{-1}\) through a magnetic field directed to the right. Determine the direction of the induced emf and the force on the rod.
Explain why a generator must have a rotating coil rather than a stationary coil in a uniform magnetic field.