Magnetostatics- Introduction And Biot Savart Law

Magnetostatics- Introduction And Biot Savart Law - Example 2

Magnetostatics is the study of the magnetic effects of electric currents.
Biot-Savart law is used to determine the magnetic field produced by a steady current.
The law is given by the equation: $ \mathbf{B} = \frac{{\mu_0}}{{4\pi}} \int \frac{{\mathbf{I} \times \mathbf{r}}}{{r^3}} dl $
Here, $ \mathbf{B} $ is the magnetic field, $ \mu_0 $ is the magnetic constant, $ \mathbf{I} $ is the current, $ \mathbf{r} $ is the position vector, and $ dl $ is the infinitesimal length element.
Let us solve an example to understand Biot-Savart law.

Example: Calculation of Magnetic Field Due to a Straight Wire

Consider a straight wire carrying a current of magnitude $ I $ .
We want to calculate the magnetic field at a point $ \mathbf{P} $ located at a distance $ r $ from the wire.
We can use the Biot-Savart law to determine the magnetic field.
The wire can be divided into small infinitesimal length elements $ dl $ .
The magnetic field produced by each element $ dl $ at point $ \mathbf{P} $ is given by: $ d\mathbf{B} = \frac{{\mu_0}}{{4\pi}} \frac{{\mathbf{I} \times \mathbf{r}}}{{r^3}} dl $
The magnetic field $ \mathbf{B} $ at point $ \mathbf{P} $ can be obtained by integrating $ d\mathbf{B} $ over the entire wire.

Example: Calculation of Magnetic Field Due to a Straight Wire (Continued)

Integrating $ d\mathbf{B} $ over the entire wire, we get: $ \mathbf{B} = \frac{{\mu_0}}{{4\pi}} \int \frac{{\mathbf{I} \times \mathbf{r}}}{{r^3}} dl $
Since the wire is straight, $ \mathbf{I} $ and $ \mathbf{r} $ are parallel.
Therefore, $ \mathbf{I} \times \mathbf{r} = 0 $ and the cross product term becomes zero.
Simplifying the equation, we obtain: $ \mathbf{B} = \frac{{\mu_0 I}}{{4\pi}} \int \frac{{dl}}{{r^2}} $
The integration over $ dl $ gives us the length of the wire, denoted by $ L $ .
Substituting this value, we get: $ \mathbf{B} = \frac{{\mu_0 I}}{{4\pi r^2}} \int dl = \frac{{\mu_0 I L}}{{4\pi r^2}} $

Example: Calculation of Magnetic Field Due to a Straight Wire (Continued)

Finally, the magnitude of the magnetic field at point $ \mathbf{P} $ is given by: $ B = \frac{{\mu_0 I}}{{4\pi r^2}} $
This equation represents the magnetic field produced by a straight wire at a point located at a distance $ r $ from the wire.
The direction of the magnetic field can be determined using the right-hand rule.
Wrap your right-hand fingers around the wire in the direction of the current.
The direction in which the thumb points gives the direction of the magnetic field.
The magnetic field lines form concentric circles around the wire.

Magnetic Field Due to a Current Loop

Now let’s consider a circular loop carrying a current of magnitude $ I $ .
We want to calculate the magnetic field at a point $ \mathbf{P} $ located on the axis of the loop.
The magnetic field at $ \mathbf{P} $ can be determined using the Biot-Savart law.
The loop can be divided into small infinitesimal current elements $ d\mathbf{I} $ .
The magnetic field produced by each element $ d\mathbf{I} $ at point $ \mathbf{P} $ is given by: $ d\mathbf{B} = \frac{{\mu_0}}{{4\pi}} \frac{{d\mathbf{I} \times \mathbf{r}}}{{r^3}} $
The magnetic field $ \mathbf{B} $ at point $ \mathbf{P} $ is obtained by integrating $ d\mathbf{B} $ over the entire loop.

Magnetic Field Due to a Current Loop (Continued)

Integrating $ d\mathbf{B} $ over the entire loop, we get: $ \mathbf{B} = \frac{{\mu_0}}{{4\pi}} \int \frac{{d\mathbf{I} \times \mathbf{r}}}{{r^3}} $
Since the loop is symmetric, the direction of $ d\mathbf{B} $ due to each element cancels out.
Thus, only the magnitude of the magnetic field at point $ \mathbf{P} $ needs to be calculated.
Simplifying the equation, we obtain: $ \mathbf{B} = \frac{{\mu_0 I}}{{4\pi}} \int \frac{{d\mathbf{l} \times \mathbf{r}}}{{r^3}} $
The integration is performed over the length $ d\mathbf{l} $ of the element.
The magnetic field $ \mathbf{B} $ is calculated by summing the contributions from all the elements of the loop.
The magnetic field lines are confined to the plane of the loop and form concentric circles around the axis.

Applications of Biot-Savart Law

The Biot-Savart law is widely applicable in various fields, including physics, engineering, and medicine.
It is used to calculate the magnetic field produced by current-carrying wires, coils, and solenoids.
The law is also used in the design of electromagnetic devices such as transformers, inductors, and motors.
Biot-Savart law plays a crucial role in understanding the behavior of charged particles in magnetic fields.
It is utilized in magnetic resonance imaging (MRI) to create detailed images of the human body.
Electromagnetic levitation systems, maglev trains, and particle accelerators rely on the principles of Biot-Savart law.
The law helps predict the behavior of plasma, a state of matter that consists of charged particles immersed in a magnetic field.
By analyzing the magnetic field produced by currents, Biot-Savart law aids in studying the Earth’s magnetic field and its impact on various phenomena.
Overall, the Biot-Savart law serves as the foundation for many applications in the field of magnetostatics.

Magnetic Field Due to a Circular Coil

A circular coil carrying a current generates a magnetic field similar to that of a bar magnet.
The magnetic field at the center of the coil is stronger compared to other points on the axis.
For a circular coil with $ N $ turns, the magnetic field at the center can be calculated using the formula: $ B = \frac{{\mu_0 N I}}{{2R}} $ where $ B $ is the magnetic field, $ \mu_0 $ is the magnetic constant, $ N $ is the number of turns, $ I $ is the current, and $ R $ is the radius of the coil.
The direction of the magnetic field can be determined using the right-hand rule.
The magnetic field lines are circular and perpendicular to the plane of the coil.
Increasing the number of turns or the current in the coil strengthens the magnetic field.
Decreasing the radius of the coil also increases the magnetic field.
This information is crucial for various applications such as electromagnets, transformers, and magnetic resonance imaging (MRI).

Magnetic Field Due to a Solenoid

A solenoid is a long, cylindrical coil with multiple turns closely packed together.
The magnetic field produced by a solenoid is similar to that of a bar magnet.
The magnetic field inside a solenoid is strong and nearly uniform.
The magnetic field outside the solenoid is weak and negligible.
The magnetic field inside a solenoid can be calculated using the formula: $ B = \mu_0 n I $ where $ B $ is the magnetic field, $ \mu_0 $ is the magnetic constant, $ n $ is the number of turns per unit length, and $ I $ is the current.
The direction of the magnetic field can be determined using the right-hand rule.
The magnetic field lines inside the solenoid are parallel and closely spaced.
Applications of solenoids include electromechanical devices, such as actuators, relays, and valves.
Solenoids are also used in medical imaging, particle accelerators, and various scientific experiments.

Ampere’s Circuital Law

Ampere’s circuital law relates the magnetic field produced by a current to the current enclosed by a closed loop.
The law states that the line integral of the magnetic field around a closed loop is equal to the product of the permeability of free space ( $ \mu_0 $ ) and the total current enclosed ( $ I_{\text{enc}} $ ): $ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} $ where $ \mathbf{B} $ is the magnetic field, $ d\mathbf{l} $ is an infinitesimal length element along the closed loop, and $ I_{\text{enc}} $ is the total current enclosed by the loop.
Ampere’s circuital law is based on experimental observations and provides a more general way to calculate magnetic fields than the Biot-Savart law.
This law is particularly useful for calculating the magnetic field in symmetric systems with high symmetry, such as long straight wires, loops, and solenoids.
Ampere’s circuital law is a fundamental principle in electromagnetism and forms the basis for Maxwell’s equations.

Magnetic Field of a Straight Wire - Infinite Length

Consider an infinitely long straight wire carrying a current $ I $ .
The magnetic field produced by the current can be determined using Ampere’s circuital law.
Assume a circular loop of radius $ r $ centered on the wire.
The magnetic field lines produced by the current are circular and concentric with the wire.
The magnetic field is constant along any circular loop of radius $ r $ around the wire.
Applying Ampere’s circuital law to the circular loop, we find: $ B \cdot 2\pi r = \mu_0 I_{\text{enc}} $ since the length of the loop is $ 2\pi r $ and the current enclosed is equal to $ I $ .
Solving for $ B $ , we obtain: $ B = \frac{{\mu_0 I}}{{2\pi r}} $
The direction of the magnetic field can be determined using the right-hand rule.

Magnetic Field of a Straight Wire - Infinite Length (Continued)

The magnitude of the magnetic field produced by an infinitely long straight wire decreases with distance from the wire.
The magnetic field is inversely proportional to the distance from the wire ( $ 1/r $ ).
Far away from the wire, the magnetic field becomes weaker and can be considered negligible.
The magnetic field lines form concentric circles around the wire, with the wire being the axis of symmetry.
The right-hand rule can be used to determine the direction of the magnetic field.
Wrap your right-hand fingers around the wire in the direction of current flow.
The direction in which your thumb points gives the direction of the magnetic field.
The magnetic field due to a straight wire is fundamental in understanding the behavior of current-carrying conductors.

Magnetic Field of a Current Loop - On-axis

Consider a circular current loop with a radius $ R $ and a current $ I $ .
We want to determine the magnetic field at a point on the axis of the loop.
The magnetic field on the axis of a current loop depends on the distance from the center of the loop ( $ z $ ).
Using Ampere’s circuital law and considering a circular loop of radius $ r $ on the axis, we find: $ B \cdot 2\pi r = \mu_0 I_{\text{enc}} $ where $ I_{\text{enc}} $ is the current enclosed by the loop.
For a current loop, the current enclosed is equal to the total current ( $ I_{\text{enc}} = I $ ).
Solving for $ B $ , we obtain: $ B = \frac{{\mu_0 I R^2}}{{2(R^2+z^2)^{3/2}}} $
The magnetic field on the axis of a current loop decreases with distance from the center.
The magnetic field is strongest at the center of the loop ( $ z = 0 $ ) and decreases as $ z $ increases.

Magnetic Field of a Current Loop - On-axis (Continued)

The magnetic field on the axis of a current loop is inversely proportional to $ (R^2+z^2)^{3/2} $ .
As the distance from the center of the loop increases, the magnetic field becomes weaker.
The magnetic field at points far away from the loop can be considered negligible.
The direction of the magnetic field on the axis of a current loop can be determined using the right-hand rule.
Point your thumb in the direction of the current in the loop.
Your fingers will curl in the direction of the magnetic field.
The magnetic field lines on the axis of a current loop are symmetric and resemble the field lines of a bar magnet.
Understanding the magnetic field of a current loop is essential in the design and analysis of electromagnets, electric motors, and other electrical devices.

Magnetic Field of a Current Loop - Off-axis

When the observation point is off the axis of a current loop, determining the magnetic field becomes more complex.
The magnetic field at a point off the axis of a current loop depends on both the lateral distance from the loop ( $ x $ ) and the vertical distance ( $ z $ ).
The magnetic field cannot be easily calculated using a simple formula.
Numerical calculations or approximations are typically used to determine the magnetic field at off-axis points.
The configuration of a current loop combined with the position of the observation point affects the magnitude and direction of the magnetic field.
The study of the magnetic field of a current loop off-axis involves advanced mathematical techniques, such as integration and vector calculus.
Engineering tools and software can be used to analyze the magnetic field for specific configurations and position scenarios.
The magnetic field produced by a current loop off-axis is essential in understanding magnetism and its applications in various fields.

Magnetic Field Inside a Toroid

A toroid is a hollow, circular-shaped object that resembles a doughnut.
It is often made of a ferromagnetic material and has a wire wound around it to form multiple turns.
The magnetic field inside a toroid is relatively strong and uniform.
Inside the toroid, the magnetic field lines are circular and concentric with the axis of the toroid.
The magnetic field inside a toroid can be calculated using Ampere’s circuital law.
Applying the law to a circular loop inside the toroid, we find: $ B \cdot 2\pi r = \mu_0 I_{\text{enc}} $ where $ r $ is the radius of the loop and $ I_{\text{enc}} $ is the current enclosed by the loop.
For a toroid, the current enclosed by the loop is the same for all loops and is equal to the total current ( $ I_{\text{enc}} = I $ ).
Thus, the magnetic field inside a toroid is given by: $ B = \frac{{\mu_0 n I}}{{2\pi r}} $ where $ n $ is the number of turns per unit length of the toroid.
The magnetic field inside a toroid is used in transformers, inductors, and other electromagnetic devices.

Magnetic Field Due to a Current Ring

Consider a circular ring carrying a current $ I $ .
We want to calculate the magnetic field at a point $ \mathbf{P} $ located on the axis of the ring.
The magnetic field at $ \mathbf{P} $ can be determined using the Biot-Savart law.
The ring can be divided into small infinitesimal current elements $ d\mathbf{I} $ .
The magnetic field produced by each element $ d\mathbf{I} $ at point $ \mathbf{P} $ is given by: $ d\mathbf{B} = \frac{{\mu_0}}{{4\pi}} \frac{{d\mathbf{I} \times \mathbf{r}}}{{r^3}} $
The magnetic field $ \mathbf{B} $ at point $ \mathbf{P} $ is obtained by integrating $ d\mathbf{B} $ over the entire ring.

Magnetic Field Due to a Current Ring (Continued)

Integrating $ d\mathbf{B} $ over the entire ring, we get: $ \mathbf{B} = \frac{{\mu_0}}{{4\pi}} \int \frac{{d\mathbf{I} \times \mathbf{r}}}{{r^3}} $
Since the ring is symmetric, the direction of $ d\mathbf{B} $ due to each element cancels out.
Thus, only the magnitude of the magnetic field at point $ \