State and use the relative directions of force, field and induced current

4.5.1 Electromagnetic Induction

Learning Objective

State and use the relative directions of the magnetic field, the induced current and the magnetic force on a conductor, and apply Faraday’s law for both motion‑induced and flux‑change‑induced e.m.f.

Key Formulae

• Motional e.m.f. (Faraday’s law for a moving conductor)

  \[

  \boxed{\mathcal{E}=B\,l\,v\sin\theta}

  \]

  • \(\mathcal{E}\) – induced e.m.f. (V)
  • \(B\) – magnetic flux density (T)
  • \(l\) – length of the conductor that lies within the field (m)
  • \(v\) – speed of the conductor relative to the field (m s\(^{-1}\))
  • \(\theta\) – angle between \(\vec v\) and \(\vec B\) (the \(\sin\theta\) term accounts for non‑perpendicular motion)

• General Faraday law (changing magnetic flux)

  \[

  \boxed{\mathcal{E}= -\dfrac{d\Phi}{dt}}

  \qquad\text{with}\qquad

  \Phi = B A \cos\alpha

  \]

  • \(\Phi\) – magnetic flux through a loop (Wb)
  • \(A\) – area of the loop (m\(^2\))
  • \(\alpha\) – angle between \(\vec B\) and the normal to the loop
  • The minus sign expresses Lenz’s law: the induced e.m.f. always opposes the change in flux.

Factors that Affect the Magnitude of the Induced e.m.f.

• Strength of the magnetic field (\(B\)).
• Length of the conductor that cuts the field (\(l\)).
• Speed of the conductor relative to the field (\(v\)).
• Angle between the direction of motion and the field (\(\theta\)); maximum when \(\theta = 90^{\circ}\).
• Number of turns (\(N\)) in a coil – the total e.m.f. is \(N\mathcal{E}\).

Direction Rules

Fleming’s Right‑Hand Rule (motional e.m.f.)

Thumb, fore‑finger and middle finger 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 conventional current (\(I\)).

Lenz’s Law

The induced current creates a magnetic field that opposes the *cause* of the induction (either a change in flux or the motion that cuts the field lines).

Fleming’s Left‑Hand Rule (force on a current‑carrying conductor)

Used when the current direction is known and the force direction is required.

• Fore‑finger – direction of magnetic field (\(\vec B\)).
• Middle finger – direction of current (\(I\)).
• Thumb – direction of magnetic force (\(\vec F\)) on the conductor.

Step‑by‑Step Procedure for Solving Problems

1. Identify the magnetic field direction (\(\vec B\)). Use \(\otimes\) (into the page) or \(\odot\) (out of the page) when drawn.
2. Determine the motion of the conductor (\(\vec v\)) or the way the flux is changing (rotation, varying field, etc.).
3. If the e.m.f. is produced by motion, apply Fleming’s right‑hand rule to obtain the direction of the induced current.
4. If the e.m.f. is produced by a changing flux, use the right‑hand grip rule for magnetic flux (curl fingers in the direction of the induced current; thumb gives the direction of the induced magnetic field) together with Lenz’s law.
5. Check the result with Lenz’s law – the induced magnetic field must oppose the original change.
6. When the question asks for the magnetic force on the conductor, apply Fleming’s left‑hand rule (or \(\vec F = I\vec l\times\vec B\)).

Typical Scenarios

Worked Example 1 – Motional e.m.f.

Problem: A straight conductor \(l = 0.20\;\text{m}\) moves at \(v = 5.0\;\text{m s}^{-1}\) to the right through a uniform magnetic field \(B = 0.30\;\text{T}\) directed into the page. Find the direction of the induced current, the magnetic force on the conductor, and the magnitude of the e.m.f.

1. \(\vec B\): into the page (\(\otimes\)).
2. \(\vec v\): to the right.
3. Fleming’s right‑hand rule → thumb (right), fore‑finger (into page), middle finger points out of the page.

   Induced current direction: out of the page.

4. Magnitude of e.m.f.:

   \[

   \mathcal{E}=B\,l\,v\sin90^{\circ}=0.30\times0.20\times5.0=0.30\;\text{V}

   \]

5. Fleming’s left‑hand rule for the force: fore‑finger (into page), middle finger (out of page), thumb points to the left.

   Force direction: opposite to the motion, as required by Lenz’s law.

Worked Example 2 – Changing Flux (rotating coil)

Problem: A single‑turn rectangular coil of area \(A = 0.040\;\text{m}^2\) rotates at a constant angular speed \(\omega = 50\;\text{rad s}^{-1}\) in a uniform magnetic field \(B = 0.25\;\text{T}\) that points out of the page. At the instant when the plane of the coil makes an angle of \(30^{\circ}\) with the field, determine the magnitude and direction of the induced e.m.f.

1. Flux: \(\Phi = B A \cos\alpha = 0.25\times0.040\times\cos30^{\circ}=8.66\times10^{-3}\;\text{Wb}\).
2. For a rotating coil, \(\displaystyle \frac{d\Phi}{dt}= -B A \omega \sin\alpha\).

   \[

   \mathcal{E}= -\frac{d\Phi}{dt}=B A \omega \sin\alpha

   =0.25\times0.040\times50\times\sin30^{\circ}=0.25\;\text{V}

   \]

3. Because \(\frac{d\Phi}{dt}<0\) (flux decreasing), the induced magnetic field must point out of the page to oppose the decrease. Using the right‑hand grip rule, the induced current is clockwise when viewed from the observer.

Summary Table

Experiment to Demonstrate Electromagnetic Induction (AO1 / AO3)

1. Apparatus: horseshoe magnet, insulated copper coil (~50 turns), galvanometer or digital multimeter, wooden rod or sliding bar to move the coil, stand and clamps.
2. Procedure:

   • Connect the coil to the galvanometer.
   • Pull the coil rapidly out of the magnetic field and note the deflection; push it back in and observe the opposite deflection.
   • Repeat with different speeds – the deflection (hence \(\mathcal{E}\)) increases with speed, confirming \(\mathcal{E}=B l v\).
   • Keep the coil stationary and vary the magnet’s field strength (move the magnet closer or farther). A deflection appears whenever the field changes, illustrating \(\mathcal{E}= -d\Phi/dt\).

3. Link to syllabus:

   • The direction of the galvanometer needle shows the direction of the induced current; it reverses when the direction of motion reverses (right‑hand rule).
   • The magnitude of the deflection is proportional to speed, field strength and number of turns – confirming the factors listed above.
   • A current is produced even without motion when the magnetic field changes, demonstrating the general Faraday law.

4. Safety note: Keep metal objects away from the strong magnet; do not allow the magnet to snap into the coil.

Common Mistakes to Avoid

• Confusing Fleming’s right‑hand rule (induced current) with the left‑hand rule (magnetic force).
• Omitting the \(\sin\theta\) factor when \(\vec v\) is not perpendicular to \(\vec B\).
• Neglecting Lenz’s law when deciding the direction of the induced current – the induced field must always oppose the cause.
• Assuming the e.m.f. is always positive; the sign in \(\mathcal{E}= -d\Phi/dt\) determines direction.
• Forgetting that a rotating coil produces a sinusoidally varying e.m.f.; the peak value is \(B A \omega\) and the instantaneous value follows \(\mathcal{E}=B A \omega \sin\omega t\).

Practice Questions

1. 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.
2. A rod of length \(0.15\;\text{m}\) moves upward at \(3.0\;\text{m s}^{-1}\) through a magnetic field directed to the right. Determine the direction of the induced e.m.f. and the magnetic force on the rod.
3. Explain why a generator must have a rotating coil rather than a stationary coil in a uniform magnetic field.
4. A single‑turn coil of area \(0.020\;\text{m}^2\) is placed in a uniform magnetic field of \(0.40\;\text{T}\) that is increasing at a rate of \(0.10\;\text{T s}^{-1}\). Calculate the magnitude of the induced e.m.f. and state the direction of the induced current.
5. In the laboratory experiment described above, the coil is moved twice as fast. By what factor does the galvanometer deflection change? Explain your answer using the relevant formula.