Ampere's Loop Theorem in College Physics

foreword

What is Ampere's loop theorem

The physical significance of Ampere's loop theorem is to describe the interaction between current and magnetic field, and how to analyze this interaction in a closed loop.

To put it simply, use the loop theorem to solve the result of the linear integration of B to any closed loop in the magnetic field, divide the path into many microelements, multiply it with the magnetic field B, and then integrate

What does Ampere's loop theorem do?

• Magnetic field generation: Ampere's loop theorem tells us that when an electric current passes through a wire or coil, a magnetic field is generated around it. The strength of this magnetic field can be calculated by loop integration. Therefore, this theorem helps us understand how electric currents generate magnetic fields.
• Magnetic field distribution: By applying the Ampere loop theorem, the magnetic field strength at various points within a closed loop can be determined. This is important for modeling and analysis of electromagnetic fields, as it allows us to predict and quantify the magnetic field strength at different locations.

understand deeper

The core idea of ​​Ampere's loop theorem can be summarized as follows:

Closed Loops: Ampere's Loop Theorem applies to a closed loop, which can be an actual circuit coil or a virtual imaginary loop.

Loop integration: In this closed loop, you can choose an arbitrary path along the path of the loop and integrate the magnetic field strength along the path. This integral is called the loop integral.

Loop integral equals total current: According to Ampere's loop theorem, the loop integral is equal to the total current through the loop multiplied by a constant μ₀, which is the magnetic permeability (μ₀). This constant is fixed in nature.

Mathematically expressed as:
∮ B dl = μ₀ * ΣI

in,

∮ represents the loop integral along the closed circuit,
B is the magnetic induction of the magnetic field,
dl represents the path element,
μ₀ is the permeability of free space, about 4π x 10^(-7) Tesla·meter/Ampere (T·m/A),
ΣI is the sum of all currents passing through the loop.
This theorem tells us that the loop integral of the magnetic field in a closed loop is equal to the total current through the loop multiplied by the magnetic permeability.

deep learning

• Use the right-hand rule to determine direction
• The left side of the formula is determined by all the magnetic fields in the space, and the right side is the current in the selected area

In the above proof, the angle between B and l is 0 degrees

In the example above, converting to radians, radians are the same