Biot-Savart Law for a Moving Point Charge
The magnetic field B at a point r due to a charge *q* moving with velocity v is given by:
```
B(r) = (μ₀ / 4π) * (q * v × r̂) / r²
```
where:
* μ₀ is the permeability of free space (approximately 4π × 10⁻⁷ T⋅m/A)
* r̂ is a unit vector pointing from the charge's position to the point r where you're calculating the field.
* r is the distance between the charge and the point r.
* × denotes the cross product.
Explanation:
* Direction: The magnetic field B is perpendicular to both the velocity vector v and the vector pointing from the charge to the observation point r. This is a direct consequence of the cross product.
* Magnitude: The strength of the magnetic field is inversely proportional to the square of the distance from the charge.
* Velocity Dependence: The magnetic field is directly proportional to the velocity of the charge. A stationary charge does not produce a magnetic field.
Important Considerations:
* This formula applies to a single point charge moving in free space.
* If there are multiple charges or if the charges are moving in a complex way, you would need to apply the Biot-Savart law to each individual charge and then superpose the resulting fields to find the total magnetic field.
Example:
Let's say you have a charge *q* moving with a velocity *v* along the x-axis. You want to find the magnetic field at a point directly above the charge on the y-axis, at a distance *d* from the charge.
1. r: The vector r points from the charge to the observation point, so r = (0, d, 0).
2. r̂: The unit vector r̂ is r / |r|, which is (0, 1, 0).
3. v: The velocity vector is v = (v, 0, 0).
4. v × r̂: The cross product is (0, 0, v).
Now, plug these values into the Biot-Savart Law:
B = (μ₀ / 4π) * (q * (0, 0, v) / d²) = (μ₀qv / 4πd²) * (0, 0, 1)
The magnetic field points in the positive z-direction, perpendicular to both the velocity and the position vector.
