Magnetic Field Due to A Current

Force per unit length: $\frac{dF}{dl} = \frac{\mu_0 i_1 i_2}{2\pi d}$

Derivation:

1. Wire 1 creates magnetic field $B_1 = \frac{\mu_0 i_1}{2\pi d}$ at the location of wire 2
2. Force on wire 2 is $F_2 = i_2 L \times B_1$
3. Force per unit length: $\frac{dF}{dl} = i_2 B_1 = i_2 \frac{\mu_0 i_1}{2\pi d} = \frac{\mu_0 i_1 i_2}{2\pi d}$

Direction:

• If currents are in same direction: wires ATTRACT each other
• If currents are in opposite directions: wires REPEL each other

Application: This principle defines the ampere. If two parallel wires 1m apart carry equal currents producing a force of $2 \times 10^{-7}$ N/m, the current is defined as 1 ampere.

Mechanism of attraction/repulsion:

For parallel currents:

1. Current in wire 1 creates magnetic field $B_1 = \frac{\mu_0 i_1}{2\pi d}$ circling around the wire
2. At wire 2's location, this field is perpendicular to wire 2
3. Force on wire 2 is $F_2 = i_2 L \times B_1$
4. Direction determined by right-hand rule: fingers along current, thumb along field, palm in force direction
5. This yields a force TOWARD wire 1 (attraction)

For antiparallel currents:

• Same mechanism, but field direction is reversed relative to second current
• Resulting force is AWAY from wire 1 (repulsion)

Connection to ampere definition:

• 1 ampere is defined as the constant current that, if maintained in two straight parallel conductors of infinite length and negligible circular cross-section placed 1 meter apart in vacuum, would produce a force of $2×10^{-7}$ newtons per meter of length
• This corresponds precisely to $\frac{dF}{dl} = \frac{\mu_0 i_1 i_2}{2\pi d} = \frac{4\pi×10^{-7}×1×1}{2\pi×1} = 2×10^{-7}$ N/m

Magnitude of magnetic field: $B = \frac{\mu_0}{4\pi} \frac{I \cdot dl \cdot \sin\theta}{r^2}$

$B = \frac{\mu_0}{4\pi} \frac{10 \text{ A} \cdot 10^{-2} \text{ m} \cdot \sin 45°}{(2 \text{ m})^2}$

$B = \frac{10^{-7} \text{ T·m/A} \cdot 10 \text{ A} \cdot 10^{-2} \text{ m} \cdot 0.707}{4 \text{ m}^2}$

$B \approx 1.8 \times 10^{-9} \text{ T}$

The approximation of treating the wire as a small element is valid because:

• The distance to the point (200 cm) is much larger than the length of the wire segment (1 cm)
• The ratio of distance to wire length is 200:1
• When this ratio exceeds about 10:1, the error from treating a finite wire as a point source becomes negligible
• This allows us to use the point-form of Biot-Savart Law rather than integrating over the length

To determine if the magnetic field goes into or out of a plane:

1. Identify the direction of current (i) and the position vector (r) from the current element to the point

2. Apply the right-hand rule to the cross product (dl × r):

   • Place your right hand along the current direction dl
   • Curl your fingers toward the position vector r
   • Your thumb points in the direction of the magnetic field

3. Conventional notation:

   • ⊗ (encircled cross) indicates field going into the plane
   • ⊙ (encircled dot) indicates field coming out of the plane

Example: For a straight current-carrying wire in a plane with current flowing upward, points to the left of the wire experience a magnetic field going into the plane, while points to the right experience a field coming out of the plane.

The magnetic field magnitude around a long, straight current-carrying wire:

$B = \frac{\mu_0 I}{2\pi r}$

Where:

• B is the magnetic field strength (in tesla, T)
• μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A)
• I is the current in the wire (in amperes, A)
• r is the perpendicular distance from the wire (in meters, m)

The field strength decreases inversely with distance from the wire.

For a wire carrying 2A at 10 cm distance: B = (4π × 10⁻⁷ T·m/A)(2A)/(2π × 0.1m) B = (4π × 10⁻⁷ T·m/A)(2A)/(0.2π m) B = 4 × 10⁻⁶ T = 4 μT

Note: Unlike electric fields that radiate outward, magnetic fields circulate around the current.

The magnetic field at an axial point is:

$B = \frac{\mu_0 i a^2}{2(a^2 + d^2)^{3/2}}$

Where:

• $\mu_0$ is the permeability of vacuum ($4\pi \times 10^{-7}$ T·m/A)
• $i$ is the current in the loop
• $a$ is the radius of the loop
• $d$ is the distance from the center of the loop to the axial point

This formula shows that:

• The field is strongest at the center ($d = 0$)
• The field decreases with increasing distance $d$
• The direction of the field is along the axis of the loop

Magnetic field strength variations around a circular current loop:

• Strongest at points inside and near the wire
• Decreases with distance from the loop (proportional to 1/r³ for far points)
• Most uniform near the center of the loop
• Varies with position along the axis

On the axis at distance x from center of a loop with radius a: $B = \frac{\mu_0 i a^2}{2(a^2 + x^2)^{3/2}}$

For points far from the loop (x >> a), this simplifies to: $B \approx \frac{\mu_0 i a^2}{2x^3} = \frac{\mu_0}{4\pi}\frac{2\pi m}{x^3}$

Where m = ia² is the magnetic dipole moment.

Example: At twice the radius distance from the center along the axis, the field strength drops to about 1/8 of its value at the center.

Ampere's Law states: The circulation of the magnetic field B along a closed, plane curve equals μ₀ times the total current crossing the area bounded by the curve:

$\oint \vec{B} \cdot \vec{dl} = \mu_0 I$

Where:

• The circulation is calculated by integrating B·dl around the entire closed loop
• Only currents crossing through the area bounded by the loop are counted
• Currents directed toward the "positive side" of the loop area are counted as positive
• Currents directed toward the "negative side" are counted as negative
• The law is valid only when electric fields remain constant inside the loop

Example: For a long straight wire carrying current I, using a circular path of radius r centered on the wire gives B(2πr) = μ₀I, leading to B = μ₀I/2πr.

The magnetic field strength (B) is inversely proportional to the distance (r) from a straight current-carrying wire:

B = μ₀i/(2πr)

This relationship exists because:

1. The magnetic field lines form concentric circles around the wire
2. As distance increases, the same field spreads over a larger circumference (2πr)
3. By conservation of energy, the field strength must decrease proportionally
4. The 1/r dependence (rather than 1/r² as in electrostatics) occurs because we're dealing with a line source rather than a point source

This inverse relationship means doubling the distance reduces the field strength by half, and can be verified experimentally by measuring the force on a test charge at various distances.

Physical significance of solenoid field distribution:

1. Inside the solenoid:

   • Uniform magnetic field parallel to axis
   • Magnitude: $B = \mu_0 ni$ (where n = turns per unit length)
   • Uniformity results from superposition of fields from all turns
   • Each turn's field reinforces others in the interior region

2. Outside the solenoid:

   • Magnetic field approaches zero for long solenoids
   • Fields from opposite sides of different turns cancel each other
   • This creates a contained magnetic field region

3. Current distribution significance:

   • The helical current forms loops around the axis
   • Each current element contributes according to Biot-Savart law
   • The winding pattern creates field lines that run parallel inside
   • Mathematically equivalent to many circular current loops placed side by side

This field containment property makes solenoids useful for creating controlled magnetic fields in applications like electromagnets, MRI machines, and particle accelerators.