Describe the pattern and direction of the magnetic field due to currents in straight wires and in solenoids

4.5.3 Magnetic Effect of a Current

Objective

Describe the pattern and direction of the magnetic field produced by:

• Currents in straight wires
• Currents in solenoids (inside and outside)

1. Magnetic Field Around a Straight Current‑Carrying Conductor

1.1 Pattern of the field

• The field lines are concentric circles centred on the wire.
• All circles lie in planes that are perpendicular to the length of the wire (the direction of the current).

1.2 Direction – Right‑Hand Thumb Rule

1. Grip the wire with the right hand so that the thumb points in the direction of conventional current (positive → negative).
2. The curled fingers show the sense in which the magnetic‑field lines circulate around the wire.

1.3 Field‑strength formula (long straight conductor)

\( B = \dfrac{\mu_{0}\,I}{2\pi r} \)

• \(\mu_{0}=4\pi\times10^{-7}\ \text{T·m·A}^{-1}\) (permeability of free space)
• \(I\) – current (A)
• \(r\) – radial distance from the centre of the wire (m)
• Units: \(B\) is measured in tesla (T = N A⁻¹ m⁻¹).

1.4 Qualitative dependence on current and distance

• Linear with current: Doubling \(I\) doubles \(B\) at a given \(r\).
• Inverse with distance: \(B\) falls off as \(1/r\); the farther from the wire, the weaker the field.
• Reversing the direction of the current reverses the sense of the circular field lines.

1.5 Simple demonstration (practical skill)

1.6 Quantitative extension (optional)

Use a Hall‑probe or a smartphone magnetometer to measure \(B\) at known distances. Compare the measured values with the theoretical prediction from the formula above and discuss any discrepancies (e.g., finite length of the wire, measurement uncertainty).

2. Magnetic Field of a Solenoid

2.1 Pattern of the field

• Inside a long solenoid: Field lines are parallel, straight and nearly uniform, running from the solenoid’s South pole toward its North pole.
• Outside the solenoid: The field is much weaker and the lines spread out radially (divergent), resembling the pattern of a bar‑magnet’s external field.

2.2 Direction – Right‑Hand Grip Rule

1. Wrap the fingers of the right hand around the coil in the direction of the current flowing through the turns.
2. The extended thumb points in the direction of the magnetic field inside the solenoid (from South to North).

2.3 Field‑strength formula (ideal long solenoid)

\( B = \mu_{0}\,n\,I \)

• \(n\) – number of turns per metre (turns m⁻¹)
• \(I\) – current through each turn (A)
• Assumes the solenoid is sufficiently long that edge effects are negligible.
• Units: \(B\) in tesla (T).

2.4 Qualitative dependence on current

• Linear with current: Doubling \(I\) doubles the field inside the solenoid.
• Reversing the current reverses the direction of the field (the North and South poles swap).
• Within the central region the field is essentially independent of position; only near the ends does it fall off (edge effects).

2.5 Edge‑effect reminder

The expression \(B = \mu_{0} n I\) is an approximation. Near the ends of a real solenoid the field lines spread out and the magnitude drops gradually to the external value.

2.6 Simple demonstration (practical skill)

2.7 Real‑world example

Electromagnets in door‑bells, electric bells, and magnetic locks are essentially solenoids. When current flows, the solenoid’s interior field becomes strong enough to attract a ferromagnetic armature, producing the audible “click”.

3. Comparison of Straight‑Wire and Solenoid Fields

4. Key Points to Remember

• The magnetic field produced by a current is always perpendicular to the direction of the current.
• Use the right‑hand thumb rule for a single straight conductor and the right‑hand grip rule for a solenoid.
• For a long straight wire, \(B\) falls off as \(1/r\); for a long solenoid the interior field is essentially uniform and independent of position (away from the ends).
• Both situations show a linear relationship with current: doubling \(I\) doubles \(B\); reversing \(I\) reverses the field direction.
• Remember the solenoid’s interior field points from the South pole toward the North pole.
• The formulae assume ideal conditions (infinitely long wire, infinitely long solenoid). Real coils exhibit edge effects and a slight reduction of \(B\) near the ends.
• Safety first: use low‑voltage sources, avoid short circuits, and keep ferromagnetic objects away from live conductors.

5. Sample Questions & Brief Answers

1. Straight wire: A wire carries \(I = 5\ \text{A}\). Find the magnetic field 5 cm from the wire.
   

   Solution: \(B = \dfrac{\mu_{0} I}{2\pi r}= \dfrac{4\pi\times10^{-7}\times5}{2\pi\times0.05}= 2.0\times10^{-5}\ \text{T}\).

2. Solenoid: A solenoid has 800 turns and a length of \(0.40\ \text{m}\). A current of \(2\ \text{A}\) flows. Find the magnetic field inside.
   

   Solution: \(n = \dfrac{800}{0.40}=2000\ \text{turns·m}^{-1}\).
   

   \(B = \mu_{0} n I = 4\pi\times10^{-7}\times2000\times2 = 5.0\times10^{-3}\ \text{T}\).

3. Direction (right‑hand rule): Determine the direction of the magnetic field at a point 3 cm to the right of a vertical wire carrying upward current.
   

   Answer: Point the right‑hand thumb upward. The fingers curl into the page on the right side, so the field at the point is into the page.

4. Current variation (solenoid): If the current in the solenoid of part 2 is increased to \(4\ \text{A}\), what happens to the field?
   

   Answer: The field doubles to \(1.0\times10^{-2}\ \text{T}\) because \(B\propto I\).

5. Reversing current (straight wire): What is the effect on the field pattern if the direction of current in the straight wire is reversed?
   

   Answer: The sense of the circular field lines is reversed; at any point the field direction is opposite to that before reversal.