Magnetostatics- Introduction And Biot Savart Law

Magnetostatics is the study of magnetic fields produced by steady electric currents or permanent magnets.
It deals with the behavior of magnetic fields in the absence of changing electric fields.
In this lecture, we will explore the Biot-Savart Law, which allows us to calculate the magnetic field produced by a current-carrying wire.

Biot-Savart Law

The Biot-Savart Law gives the magnetic field at a point in space due to a current element.
It states that the magnetic field at a distance r from a current-carrying element of length dl is directly proportional to the product of the current, the length element, and the sine of the angle between the position vector r and the length element dl.
The equation for the Biot-Savart Law is: B = (μ₀/4π) * (I * dl × r) / r^2 where:
- B is the magnetic field,
- μ₀ is the permeability of free space (4π × 10⁻⁷ Tm/A),
- I is the current in the wire,
- dl is the length element of the current-carrying wire,
- r is the position vector from the current element to the point where the magnetic field is measured.

Applications of Biot-Savart Law

The Biot-Savart Law can be used to find the magnetic field due to various current-carrying configurations:
- Straight wire
- Circular loop
- Solenoid
- Toroid
It is an important tool for calculating magnetic fields in magnetostatics problems.

Magnetic Field due to a Straight Wire

Consider a straight wire carrying current I.
To find the magnetic field at a point P, a distance r away from the wire, using the Biot-Savart Law, we divide the wire into small length elements dl.
The magnitude of the magnetic field dB produced by a small length element dl at point P is given by: dB = (μ₀/4π) * (I * dl * sinθ) / r² where θ is the angle between the line drawn from the wire element to the point P and the wire itself.

Magnetic Field due to a Straight Wire (contd.)

The direction of the magnetic field dB at point P due to a small length element dl is perpendicular to both dl and r, following the right-hand rule.
By integrating all the contributions of dl along the wire, we can obtain the total magnetic field B at point P.
The equation for the magnetic field due to a straight wire is: B = (μ₀/4π) * (I * L * sinθ) / r² where L is the total length of the wire.

Magnetic Field due to a Circular Loop

Consider a circular loop of radius R carrying current I.
To find the magnetic field at a point P along the axis of the loop, a distance z away from the center, we divide the loop into small current elements dl.
The magnitude of the magnetic field dB produced by a small current element dl at point P is given by: dB = (μ₀/4π) * (I * dl * sinθ) / r² where r is the distance from the current element to point P and θ is the angle between r and the axis of the loop.

Magnetic Field due to a Circular Loop (contd.)

The magnetic field dB at point P due to a current element dl is perpendicular to both dl and r, following the right-hand rule.
By integrating all the contributions of dl around the loop, we can obtain the total magnetic field B at point P.
The equation for the magnetic field due to a circular loop at its axis is: B = (μ₀/4π) * (I * R²) / (2 * R² + z²)^(3/2)

Magnetic Field due to a Solenoid

A solenoid is a tightly wound coil of wire.
It consists of many turns of wire with current flowing through it.
To find the magnetic field inside a solenoid, we consider the contribution from the individual current loops formed by the turns of the solenoid.

Magnetic Field due to a Solenoid (contd.)

The magnetic field inside an ideal solenoid is uniform and parallel to the axis of the solenoid.
The magnitude of the magnetic field inside the solenoid is given by: B = (μ₀/4π) * (N * I) where N is the number of turns per unit length.
The magnetic field outside the solenoid is negligible.

Magnetostatics- Introduction And Biot Savart Law - Problem for solving

A long straight wire carries a current of 4 A. Calculate the magnetic field produced at a point situated 20 cm away from the wire. Solution:
Given: Current (I) = 4 A, Distance from the wire (r) = 20 cm = 0.2 m
Using the Biot-Savart Law, the magnetic field at a point due to a long straight wire is given by:
- B = (μ₀/4π) * (I/ r)
Plugging in the values, we get:
- B = (4π × 10⁻⁷ Tm/A * 4 A) / (4π × 0.2 m)
- B = 10⁻⁶ T

Applications of Biot-Savart Law

The Biot-Savart Law is used to calculate the magnetic field produced by various current-carrying configurations.
Some applications include:
- Designing and analyzing electromagnets.
- Understanding the behavior of magnetic fields around current-carrying wires.
- Calculating the magnetic field inside and outside of solenoids and toroids.
- Investigating the behavior of magnetic fields around circular loops and straight wires.

Magnetic Field due to a Circular Loop - Example

Consider a circular loop with a radius of 10 cm carrying a current of 2 A. Find the magnetic field at the center of the loop. Solution:
Given: Radius (R) = 10 cm = 0.1 m, Current (I) = 2 A
To find the magnetic field at the center of the loop, we use the equation:
- B = (μ₀/4π) * (I * R²) / (2 * R²)
Plugging in the values, we get:
- B = (4π × 10⁻⁷ Tm/A * 2 A * (0.1 m)²) / (2 * (0.1 m)²)
- B = 10⁻⁷ T

Magnetic Field due to a Solenoid - Example

A solenoid has 500 turns per meter and carries a current of 6 A. Calculate the magnetic field inside the solenoid. Solution:
Given: Number of turns per unit length (N) = 500 turns/m, Current (I) = 6 A
To find the magnetic field inside the solenoid, we use the equation:
- B = (μ₀/4π) * (N * I)
Plugging in the values, we get:
- B = (4π × 10⁻⁷ Tm/A * 500 turns/m * 6 A)
- B = 6 × 10⁻⁴ T

Magnetic Field due to a Toroid

A toroid is a donut-shaped object with current flowing through a wire wound around the circumference.
The magnetic field inside a toroid is usually assumed to be uniform and parallel to the axis of the toroid.
The magnetic field outside the toroid is negligible. Properties of a toroid:
Inner radius (r1)
Outer radius (r2)
Number of turns (N)
Current (I)

Magnetic Field due to a Toroid (contd.)

The magnetic field inside a toroid can be calculated using the equation:
- B = (μ₀/2π) * (N * I) / (r1 + r2) Example:
Consider a toroid with an inner radius of 4 cm, an outer radius of 6 cm, 500 turns, and a current of 3 A. Find the magnetic field inside the toroid. Solution:
Given: Inner radius (r1) = 4 cm = 0.04 m, Outer radius (r2) = 6 cm = 0.06 m, Number of turns (N) = 500 turns, Current (I) = 3 A
Plugging in the values into the equation, we get:
- B = (4π × 10⁻⁷ Tm/A * 500 turns * 3 A) / (0.04 m + 0.06 m)
- B = 1.2 × 10⁻⁴ T

Magnetic Field due to a Current Loop

A current loop is a closed path formed by a current-carrying wire.
The magnetic field due to a current loop depends on its shape and orientation. Equation for the magnetic field at a point on the axis of a current loop:
B = (μ₀/4π) * (I * Area) / (z² + R²)^(3/2) where:
I is the current in the loop,
Area is the area enclosed by the loop,
z is the distance from the loop along its axis,
R is the radius of the loop.

Magnetic Field due to a Current Loop (contd.)

Example:

Consider a circular current loop with a radius of 5 cm carrying a current of 4 A. Find the magnetic field at a point along the axis of the loop, 10 cm away from the center. Solution:
Given: Radius (R) = 5 cm = 0.05 m, Current (I) = 4 A, Distance from the center (z) = 10 cm = 0.1 m
To find the magnetic field at the given point, we use the equation:
- B = (μ₀/4π) * (I * π * R²) / (0.1² + 0.05²)^(3/2)
- B ≈ 1.54 × 10⁻⁶ T

Magnetic Field due to a Current Loop (contd.)

The magnetic field is maximum at the center of a current loop and decreases as you move away from the center along the axis.
The magnetic field approaches zero at large distances compared to the radius of the loop. Properties of a current loop:
Radius (R)
Current (I)
Distance along the axis (z) Equation for the magnetic field along the axis of a current loop:
B = (μ₀/2R) * (I * π * R²)

Magnetic Field due to a Current Loop (contd.)

Example:

Consider a circular current loop with a radius of 8 cm carrying a current of 3 A. Find the magnetic field at a point along the axis of the loop, 20 cm away from the center. Solution:
Given: Radius (R) = 8 cm = 0.08 m, Current (I) = 3 A, Distance from the center (z) = 20 cm = 0.2 m
To find the magnetic field at the given point, we use the equation:
- B = (μ₀/16) * (I * π * R²) / (R)
- B ≈ 1.18 × 10⁻⁵ T

Electromagnets

Electromagnets are temporary magnets created by wrapping a coil of wire (usually insulated copper wire) around a ferromagnetic core.
When an electric current flows through the coil, it produces a magnetic field.
The strength of the magnetic field can be controlled by changing the magnitude of the current or by adjusting the number of turns in the coil.
Electromagnets have various applications, including in motors, generators, speakers, and magnetic resonance imaging (MRI) machines.

Designing Electromagnets

The strength of an electromagnet can be increased by:
- Increasing the current flowing through the coil.
- Increasing the number of turns in the coil.
- Using a ferromagnetic core, such as iron or steel, to concentrate the magnetic field.
The equation for the magnetic field produced by an electromagnet is similar to that of a solenoid:
- B = (μ₀/4π) * (N * I)
The magnetic field produced is directly proportional to the number of turns in the coil and the current flowing through it.

Magnetic Field due to a Current-carrying Conductor

The magnetic field generated by a current-carrying conductor is concentrated around the conductor and forms concentric circles.
The direction of the magnetic field can be determined using the right-hand rule: point your thumb in the direction of the current and your fingers curl in the direction of the magnetic field.
The magnetic field produced decreases with distance from the conductor, following an inverse square law.
The magnetic field lines around a straight wire are parallel, while those around a loop form concentric circles.

Calculation of Magnetic Field

The magnetic field produced by a current-carrying conductor at a point can be calculated using the Biot-Savart Law.
The Biot-Savart Law states that the magnetic field at a point due to a current element is directly proportional to the product of the current, the length element, and the sine of the angle between the position vector and the length element.
By integrating the contributions of all current elements along the conductor, we can find the total magnetic field at a point.

Ampere’s Circuital Law

Ampere’s Circuital Law relates the magnetic field around a closed loop to the current passing through the loop.
The law states that the line integral of the magnetic field around a closed loop is equal to the μ₀ times the total current passing through the loop.
This law is a consequence of the fact that magnetic fields are produced by electric currents.

Magnetic Flux

Magnetic flux is a measure of the number of magnetic field lines passing through a given area.
It is denoted by the symbol Φ and is defined as the dot product of the magnetic field B and the area vector A:
- Φ = B · A = B * A * cosθ
The SI unit of magnetic flux is the Weber (Wb), which is equal to one tesla-meter squared (Tm²).
The magnetic flux through a closed surface is always zero, as the number of field lines entering the surface is equal to the number of field lines leaving it.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law of Electromagnetic Induction states that a change in magnetic flux through a closed loop of wire induces an electromotive force (emf) or voltage in the wire.
The induced emf is directly proportional to the rate of change of magnetic flux and is given by:
- ε = -dΦ/dt
The negative sign indicates that the induced emf opposes the change in magnetic flux.
This law forms the basis for the operation of generators and transformers.

Lenz’s Law

Lenz’s Law, which is a consequence of Faraday’s Law, states that the direction of the induced current or emf is such that it opposes the change that produced it.
This means that when the magnetic flux through a loop increases, the induced current will create a magnetic field that opposes the increase in flux.
Similarly, when the magnetic flux through a loop decreases, the induced current will create a magnetic field that opposes the decrease in flux.
Lenz’s Law is a manifestation of the principle of conservation of energy.

Applications of Faraday’s Law

Faraday’s Law of Electromagnetic Induction has numerous practical applications, such as:
- Electric power generation: Generators convert mechanical energy into electrical energy by rotating a coil of wire in a magnetic field.
- Transformers: Transformers use the principle of electromagnetic induction to change the voltage of alternating current (AC) power for transmission and distribution.
- Magnetic sensors: Various sensors, such as Hall effect sensors, use electromagnetic induction to measure magnetic fields.
- Induction cooktops: Induction cooktops use magnetic fields generated by alternating currents to heat induction-compatible cookware.

Summary

Magnetostatics deals with the behavior of magnetic fields produced by steady electric currents or permanent magnets.
The Biot-Savart Law allows us to calculate the magnetic field at a point due to a current-carrying wire or conducting loop.
Electromagnets are temporary magnets created by flowing electric current through a coil.
Ampere’s Circuital Law relates the magnetic field around a closed loop to the current passing through the loop.
Faraday’s Law of Electromagnetic Induction states that a change in magnetic flux through a closed loop induces an emf or current.
Lenz’s Law states that the induced current or emf opposes the change that produced it.
These laws and principles have widespread applications in various technologies and everyday life.