Magnetic Fields and Biot-Savart Law

Magnetic Field Concepts

• Discussion of key topics: magnetic lines of force, the force experienced by a moving charge, a conductor, and a current-carrying conductor in a magnetic field.
• Focus on the most important aspects of these topics.

Magnetic Field Development

• Magnetic field develops at a point, and everything is related to this development.
• Small magnetic field is developed at point P.
• This small magnetic field is directly proportional to the sine of the angle (sin θ).

Biot-Savart Law

• The magnetic field strength is inversely proportional to the square of the distance (1/r^2).
• Magnetic field is directly proportional to IDL sin θ / r^2.
• Introduction of a constant, K, to remove the proportionality sign.
• K = μ₀ / 4π (where μ₀ is the permeability of free space).
• Formula for the small magnetic field dB = (μ₀ / 4π) * (IDL sin θ / r^2).
• IDL represents the current element (I) times the length element (DL).

Calculating Total Magnetic Field

• Calculation of total magnetic field involves summing up (integrating) the small magnetic fields (dB) over the entire length of the conductor.

Application: Magnetic Field at the Center of a Circular Coil

• Focus shifts to finding the total magnetic field at the center of a circular coil.
• The circular coil has a radius R, and a small element DL is considered.
• The total length of the coil is 2πR.
• dB = μ₀ / 4π * IDL / R^2 (since sin θ = sin 90° = 1).
• Integrating dB over the entire loop gives the total magnetic field B.
• Total magnetic field B = ∫dB = ∫ (μ₀ / 4π) * (IDL / R^2) from 0 to 2πR.
• B = (μ₀ * I) / (2R) after integration.

Magnetic Field with Multiple Turns

• If the circular coil has N number of turns, the total magnetic field is B = N * (μ₀ * I) / (2R).

Magnetic Field at the Axis of a Circular Coil

• Discussion shifts to the magnetic field at a point on the axis of a circular coil.
• The axis is a line passing through the center of the circular coil.
• The overall goal is to find the value of the total magnetic field (B).
• Focuses on the magnetic field produced by the small elements at Point 1 (P1)
• dB1 (magnetic field at P1) is calculated considering the small element.
• Magnetic field produced by a small element can be calculated due to a small element at point C if needed

Equations

• dB = μ₀ / 4π * (IDL sin α) / r^2 where α = 90 degrees.
• dB = μ₀ / 4π * (IDL) / (x^2 + R^2)

Vertical Components

• The vertical components DB Cos(Theta) cancel each other out.
• This happens because they are equal in magnitude but opposite in direction
• Only the horizontal components (dB sin θ) contribute to the net magnetic field (B).
• Equation for Total Magnetic Field: B = ∫dB sin θ

Calculating Total Magnetic Field

• B = ∫ (μ₀ / 4π) * (IDL / (x^2 + R^2)) * (R / (x^2 + R^2)^(1/2))
• B = (μ₀ * I * R) / (4π * (x^2 + R^2)^(3/2)) * ∫DL
• Integrating DL from 0 to 2πR gives B = (μ₀ * I * R^2) / (2 * (x^2 + R^2)^(3/2))

General Equation

• B = (μ₀ * N * I * R^2) / (2 * (x^2 + R^2)^(3/2)) for N number of turns.

Biot-Savart Law Reminder

• States dB = (μ₀ / 4π) * (I DL sin θ) / r^2.
• θ is the angle between the current element and radius vector.

Infinitely Long, Straight Conductor

• Consideration of a very long, straight conductor.
• Relates angles involving θ and Φ such that θ + Φ + 90° = 180°.

Infinitely Long Straight Wire

• The objective is to find dB = (μ₀/4π) (I dL sin θ) /r^2
• dL = r sec^2(Φ) dΦ due to derivative result
• sin(θ) = cos(Φ)
• dB = (μ₀ I cos(Φ) dΦ) / (4πr)

Infinitely Long Wire (Continued)

• Now take integral of dB from -90 to 90
• B = (μ₀I) / (2πr)
• The direction of magnetic field follows same direction
• Overview of other relationships, where full and white solenoids have virtually zero current