Magnetic Field Due to Current Carrying Element

The relationship between electricity and magnetism is a fundamental aspect of electromagnetism. A current carrying conductor generates a magnetic field around it, which can be determined by various laws and principles. One of the most important principles used to calculate the magnetic field due to a current carrying element is the Biot-Savart Law.

Biot-Savart Law

The Biot-Savart Law relates the magnetic field B to the magnitude, direction, length, and proximity of the electric current. The law is given by the formula:

$$ d\vec{B} = \frac{\mu_0}{4\pi} \frac{Id\vec{s} \times \hat{r}}{r^2} $$

where:

• d\vec{B} is the infinitesimal magnetic field produced at a point in space by the infinitesimal current element Id\vec{s}.
• I is the current passing through the element.
• d\vec{s} is the infinitesimal length vector of the current element in the direction of current.
• \hat{r} is the unit vector from the current element to the point at which the magnetic field is being calculated.
• r is the distance between the current element and the point at which the magnetic field is being calculated.
• \mu_0 is the permeability of free space, and its value is 4\pi x 10^{-7} T·m/A (Tesla meter per Ampere).

Important Points

• The direction of d\vec{B} is perpendicular to both d\vec{s} and \hat{r}, as given by the right-hand rule.
• The magnitude of the magnetic field decreases with the square of the distance from the current element.
• The Biot-Savart Law is applicable to any current distribution, provided the current is steady (i.e., it does not change with time).

Ampere's Circuital Law

For a long, straight conductor carrying a steady current, Ampere's Circuital Law provides a simpler way to calculate the magnetic field:

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

where B is the magnetic field at a distance r from the wire, and I is the current through the wire.

Important Points

• Ampere's Law is particularly useful for calculating the magnetic field in symmetrical situations, such as long straight wires, solenoids, and toroids.
• The direction of the magnetic field is given by the right-hand grip rule: if the thumb of the right hand points in the direction of the current, the fingers curl in the direction of the magnetic field lines.

Differences Between Biot-Savart Law and Ampere's Circuital Law

Biot-Savart Law	Ampere's Circuital Law
It is a general law that can be applied to any current configuration.	It is usually applied to symmetrical situations with steady currents.
The calculation involves an integral over the current distribution.	The calculation often involves applying the law to an Amperian loop.
It gives the magnetic field at a point due to a differential current element.	It gives the magnetic field due to an entire closed loop of current.
More complex to use in non-symmetrical situations.	Simpler to use in symmetrical situations.

Example: Magnetic Field of a Straight Current-Carrying Wire

Consider a long straight wire carrying a current I. To find the magnetic field at a point P located at a distance r from the wire, we can use Ampere's Law:

1. The current I flows through the wire.
2. Point P is at a distance r from the wire.
3. Using Ampere's Law, the magnetic field B at point P is given by:

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

This formula tells us that the magnetic field is inversely proportional to the distance from the wire and directly proportional to the current.

Conclusion

Understanding the magnetic field due to a current carrying element is crucial for the study of electromagnetism and its applications. The Biot-Savart Law and Ampere's Circuital Law are two important tools for calculating magnetic fields in various situations. These principles not only explain the behavior of magnetic fields around current-carrying conductors but also form the basis for many technological applications, including electric motors, generators, and magnetic resonance imaging (MRI).