sketch magnetic field patterns due to the currents in a long straight wire, a flat circular coil and a long solenoid

Magnetic Fields due to Currents – Cambridge IGCSE/A‑Level (9702)

Learning Objectives

• Define a magnetic field and state its SI unit (tesla, T).
• Explain the conventions used for magnetic‑field lines and magnetic flux density \( \mathbf{B} \).
• Apply the three right‑hand rules to determine the direction of \( \mathbf{B} \) for a straight wire, a circular coil and a solenoid.
• Use Fleming’s left‑hand rule to find the direction of the magnetic force on a current‑carrying conductor or on a moving charge.
• Sketch the magnetic‑field patterns produced by:

  • a long straight current‑carrying wire,
  • a flat circular coil (single loop), and
  • a long solenoid (many turns).

• Apply the quantitative expressions for field strength and magnetic force.
• Carry out simple practical investigations (compass mapping, Hall‑probe measurements) and relate observations to theory.
• Identify common misconceptions and avoid typical exam pitfalls.

1. Concept of a Magnetic Field (Syllabus 20.1)

• Definition: A magnetic field \( \mathbf{B} \) is a region of space in which a moving charge or a current‑carrying conductor experiences a magnetic force.
• Magnetic flux density (\( \mathbf{B} \)) is measured in tesla (T), where

  \[

  1\;\text{T}=1\;\frac{\text{N}}{\text{A·m}}=10^{4}\;\text{gauss (G)}.

• Field‑line conventions:

  • The tangent to a field line gives the direction of \( \mathbf{B} \) at that point.
  • Line density is proportional to the magnitude of \( \mathbf{B} \).
  • Field lines form closed loops; they never start or end in empty space.

• Ampère’s law (steady currents):

  \[

  \oint \mathbf{B}\!\cdot\!d\mathbf{l}= \mu{0}I{\text{enc}},\qquad

  \mu_{0}=4\pi\times10^{-7}\;\text{T·m·A}^{-1}.

  \]

2. Right‑Hand Rules (Syllabus 20.2)

3. Force on a Current‑Carrying Conductor (Syllabus 20.2)

• Formula:

  \[

  \mathbf{F}=BIL\sin\theta,

  \]

  where \(I\) is the current, \(L\) the length of the conductor within the field, \(\theta\) the angle between \(\mathbf{B}\) and \(\mathbf{L}\), and the direction of \(\mathbf{F}\) is given by Fleming’s left‑hand rule.

• Fleming’s left‑hand rule:

  • First finger → direction of magnetic field \( \mathbf{B} \) (from North to South).
  • Second finger → direction of conventional current \( I \).
  • Thumb → direction of the force on the conductor.

• Worked example:

  A 0.50 m straight wire carries \(I=3.0\;\text{A}\) in a uniform magnetic field \(B=2.0\times10^{-5}\;\text{T}\). The field is directed at \(30^{\circ}\) to the wire. Find the magnitude of the force.

  \[

  F = BIL\sin\theta = (2.0\times10^{-5})(3.0)(0.50)\sin30^{\circ}

  = 1.5\times10^{-5}\;\text{N}.

  \]

  Using Fleming’s left hand, the thumb points in the direction of this force.

4. Force on a Moving Charge (Syllabus 20.3)

• Formula:

  \[

  \mathbf{F}=q\mathbf{v}\times\mathbf{B}=qvB\sin\theta,

  \]

  where \(q\) is the charge, \(\mathbf{v}\) its velocity, and \(\theta\) the angle between \(\mathbf{v}\) and \(\mathbf{B}\).

• Direction is given by the right‑hand rule for the cross‑product (point fingers along \(\mathbf{v}\), curl toward \(\mathbf{B}\), thumb points in the direction of \(\mathbf{F}\) for a positive charge; opposite for a negative charge).
• Example: A proton (\(q=+1.6\times10^{-19}\) C) moves at \(2.0\times10^{6}\) m s\(^{-1}\) perpendicular to a magnetic field of 0.30 T.

  \[

  F = qvB = (1.6\times10^{-19})(2.0\times10^{6})(0.30)=9.6\times10^{-14}\;\text{N}.

  \]

  The force is directed according to the right‑hand rule.

5. Magnetic‑Field Patterns (Syllabus 20.3)

5.1 Long Straight Wire

• Direction: Concentric circles centred on the wire (right‑hand rule).
• Magnitude (at distance \(r\)):

  \[

  B=\frac{\mu_{0}I}{2\pi r}\qquad(\propto 1/r).

  \]

• Sketching tips:

  • Draw a thin vertical line for the wire.
  • Add several circles around it; use arrows to show the sense of rotation.
  • Label a representative radius \(r\) and write the formula beside the diagram.

• Practical example: Place a small compass at various distances from a current‑carrying wire and record the needle direction to map the circular pattern.

5.2 Flat Circular Coil (Single Loop)

• Direction: Field emerges perpendicularly from one face of the loop, passes through the centre, and re‑enters the opposite face (right‑hand rule).
• On‑axis field (distance \(x\) from the centre):

  \[

  B=\frac{\mu_{0}IR^{2}}{2\bigl(R^{2}+x^{2}\bigr)^{3/2}},

  \]

  where \(R\) is the coil radius.

• Key features:

  • Maximum at the centre (\(x=0\)): \(B{\text{centre}}=\mu{0}I/2R\).
  • Inside the loop the lines are dense and almost parallel; outside they spread out.
  • With many turns the field inside approaches that of a solenoid.

• Sketching tips:

  • Draw a circle for the coil and indicate the current direction (clockwise or anticlockwise).
  • From the near face draw arrows emerging perpendicular to the plane, passing through the centre, and returning on the far face.
  • Show a few curved lines outside the coil to illustrate the spreading field.

• Practical example: Use a Hall probe on the coil axis to verify the \(x\)-dependence of the formula above.

5.3 Long Solenoid (Many Turns)

• Direction: Inside the coil the field lines are parallel to the axis; outside they curve from one end to the other, giving a dipole‑like pattern.
• Ideal internal field (infinitely long):

  \[

  B=\mu_{0}nI,\qquad n=\frac{N}{L}\;( \text{turns per metre}).

  \]

• Characteristics:

  • Field is essentially uniform across the cross‑section.
  • Ends behave like the north and south poles of a bar magnet.
  • External field is weak; for a finite solenoid it falls roughly as \(1/r^{3}\) outside the ends.

• Sketching tips:

  • Draw a cylinder to represent the coil.
  • Inside, fill with dense, parallel arrows pointing from the South to the North pole (according to the chosen current direction).
  • At each end, draw curved lines exiting/entering the coil and label the poles.
  • Write the formula \(B=\mu_{0}nI\) beside the diagram.

• Practical example: Measure the field inside a long solenoid with a Hall probe and compare with the theoretical value \(\mu_{0}nI\).

6. Comparison of the Three Configurations

7. Practical Skills (Syllabus 20.4 & 20.5)

• Mapping field lines: Place a small compass at regular points around the conductor; draw the direction indicated by the needle to produce a field‑line map.
• Using a Hall probe: Record the magnitude of \( \mathbf{B} \) at known distances; compare with the theoretical expressions for each configuration.
• Verifying the right‑hand rule: Set up a current‑carrying wire with a known direction; use a compass or magnetic needle to confirm the predicted circular pattern.
• Safety notes: Limit the current with a resistor or low‑voltage supply, keep metallic objects away from strong magnetic fields, and never place the Hall probe near high currents without proper shielding.

8. Example Exam Question (Cambridge 9702 – 20.3)

Question: A long straight wire carries a current of 5.0 A. A circular coil of radius 0.10 m, lying in a plane perpendicular to the wire, carries a current of 2.0 A in the clockwise direction when viewed from the wire. Sketch the magnetic‑field pattern in the region where the coil and wire overlap and state the direction of the net field at a point 2 cm to the right of the wire on the coil’s axis.

Solution outline:

1. Apply the right‑hand rule for the wire → field circles clockwise when looking towards the observer; at the specified point the field is into the page.
2. For the coil, use the right‑hand rule for a loop (clockwise current → field into the page at the centre). Hence the coil also contributes into the page at the point.
3. Both contributions are in the same direction, so the net field is into the page. Its magnitude is

   \[

   B{\text{wire}}=\frac{\mu{0}I}{2\pi r}

   =\frac{4\pi\times10^{-7}\times5.0}{2\pi\times0.02}=5.0\times10^{-5}\;\text{T},

   \]

   and the coil’s axial field at \(x=0.02\;\text{m}\) is

   \[

   B{\text{coil}}=\frac{\mu{0}IR^{2}}{2(R^{2}+x^{2})^{3/2}}

   =\frac{4\pi\times10^{-7}\times2.0\times0.10^{2}}{2(0.10^{2}+0.02^{2})^{3/2}}

   \approx 1.2\times10^{-5}\;\text{T}.

   \]

   Net \(B\approx6.2\times10^{-5}\;\text{T}\) into the page.

4. Sketch: draw the circular field lines around the wire, the axial lines emerging from the near face of the coil, and indicate the combined direction at the marked point.

9. Common Misconceptions to Avoid

• “Magnetic field lines start or end on charges.” – They always form closed loops.
• Confusing the direction of current with the direction of the magnetic field; always apply the right‑hand rule first, then label the current.
• Assuming a short solenoid has a perfectly uniform field – edge effects become significant when the length is comparable to the radius.
• Using the straight‑wire formula inside a coil; the geometry changes the distance dependence.
• Applying Fleming’s left‑hand rule to a moving charge – that rule is for forces on current‑carrying conductors; for charges use the right‑hand rule for the cross product.

10. Key Points to Remember (Revision Checklist)

1. Definition of \( \mathbf{B} \) and its unit (tesla, 1 T = 10⁴ G).
2. Field‑line conventions: direction = tangent, density = magnitude, closed loops.
3. Three right‑hand rules for wire, coil, and solenoid.
4. Fleming’s left‑hand rule for the magnetic force on a conductor.
5. Fundamental formulas:

   • Wire: \( B=\mu_{0}I/2\pi r \)
   • Loop (on axis): \( B=\mu_{0}IR^{2}/[2(R^{2}+x^{2})^{3/2}] \)
   • Solenoid (ideal): \( B=\mu_{0}nI \)
   • Force on conductor: \( F=BIL\sin\theta \)
   • Force on charge: \( F=qvB\sin\theta \)

6. Sketching guidelines:

   • Concentric circles for a straight wire.
   • Emerging/returning loops for a coil.
   • Dense parallel arrows inside a solenoid, curved lines outside, with labelled poles.

7. Link theory to practice – compass mapping, Hall‑probe measurements, safety precautions.
8. Read exam questions carefully: they may ask for direction, magnitude, a sketch, or a combination.