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.

11. 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.

12. 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).

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. 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 $ \