For a coil of radius $R$ carrying current $I$, the field at a point on its axis $x$:

$B_x = \frac{\mu_0 IR^2 N}{2(x^2 + R^2)^{3/2}}$

At center of loop:

$B = \frac{\mu_0 IN}{2R}$

For large distances:

$B = \frac{\mu_0 INR^2}{2x^3}$

Magnetic moment:

$\mu = I\pi R^2$

So the field becomes:

$B = \frac{\mu_0}{2\pi} \cdot \frac{\mu}{x^3}$

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

$\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi a}$

Definition of Ampere:

$\frac{F}{l} = 2 \times 10^{-7} \, \text{N/m} \Rightarrow 1 \, \text{A}$

$\oint \vec{B} \cdot \vec{dl} = \mu_0 I_{\text{enclosed}}$

Application Steps:

1. Check symmetry (cylindrical, toroidal, etc.)
2. Use right-hand rule for $\vec{B}$
3. Choose appropriate Amperian loop
4. Evaluate $\vec{B} \cdot \vec{dl}$
5. Solve for $B$

• Field inside a long solenoid
• Field inside a toroid
• Field around a straight conductor

• Use Right-hand Grip Rule for magnetic field direction
• Use Fleming's Left Hand Rule for magnetic force direction
• Same direction currents → attract
• Opposite direction currents → repel