# Ampere's Circuit Law

André-Marie Ampère:

André-Marie Ampère was a scientist who experimented with the forces that act on current-carrying wires. The experiment was done in the late 1820s, around the same time Faraday was working on Faraday's law. Little did Faraday and Ampere know that Maxwell himself would combine their work four years later.

Ampere's Law:

According to Ampere's law, magnetic fields are related to the electric current generated in them. The law defines a magnetic field associated with a certain current, or vice versa, if the electric field does not change with time.

Ampere's law can be expressed as:

"The magnetic field produced by an electric current is proportional to the magnitude of that electric current by a proportionality constant equal to the permeability of free space."

The equation that explains Ampere's law, which is Maxwell's final equation, is given below:

Ampere's Circuit Law:

Ampere's circuit law can be written as the line integral of the magnetic field surrounding a closed circuit is equal to the algebraic sum of the currents through the circuit.

Ampere's Circuit Law

It is assumed that there is a current I in the conductor, then this current creates a magnetic field that surrounds the conductor.

The left side of the equation describes that if the wire is surrounded by an imaginary path and a magnetic field is added at each point,  it is numerically equal to the current surrounded by that path, indicated by Ienc.

Determining Magnetic Field by Ampere’s Law (Example):

Suppose you have a wire long enough to carry a continuous current of I amperes. How do you determine the magnetic field surrounding the wire at any distance r from the wire?

The image below (image 1) shows a long wire carrying a current in amperes. We need to find the magnitude of the magnetic field at a distance r. Therefore, we draw the wire marked with a blue dashed line around the imaginary path on the right of the image. Determination of magnetic field (according to Ampere's law)

Figure 1

According to the second equation, if the  field is integrated along the blue path,  it must equal  the closed current  I.

The magnetic field does not change with distance r due to symmetry. The length of the path (in blue) in Figure 1 is equal to the circumference of the circle 2πr.

If a constant value H is added to the field, the  left side of the equation looks like this:

Ampere's Law

We have calculated the magnitude of H. Since r is arbitrary, the value of the field H is known.  The magnitude of the magnetic field decreases as you move wider according to Eq. Therefore, Ampere's law can be applied to calculate the magnitude of the field around the wire. The field H is a vector field indicating that each region has a direction and a magnitude. The direction of the field is tangent at each point of the imaginary loops, as shown in Figure 2, and the right rule finds the direction of the magnetic field.

Applications of Ampere's Law

Ampere's law is used:

Determine the magnetic induction due to a long current-carrying wire.

Determine the magnetic field inside the toroid.

Determine the magnetic field produced by a long current-carrying cylinder.

Magnetic Field Due to a Cylindrical Wire:

Magnetic field due to a cylindrical wire is obtained by the application of Ampere’s law.

Outside the Cylinder:

In all above cases magnetic field outside the wire at P, ∫B̄.dl̄̄ = µ₀I  B ∫dl = µ₀i.

B x 2πr = µ₀i  Bout = µ₀i/ 2πr

In all the above cases, B surface = µ₀i/ 2πR

Inside the hollow cylinder: Magnetic field inside the hollow cylinder is zero.

Inside the solid cylinder: Current enclosed by loop (I) is lesser than the total current.

Inside the solid cylinder

Current density is uniform, i.e. J = J i’ = i x (A’ x A) = i (r²/R²), hence at inside point ∫B̄in.dl̄ = µ₀’ B = µ₀ir/ 2πR².

Inside the thick portion of the hollow cylinder: Current enclosed by the loop is given by as, i’ = i x (A’/ A) = i x [(r² – R₁²)/ (R₂² – R₁²)]

Inside the thick portion of hollow cylinder

Solenoid

Electromagnets

Electromagnets are magnets with a wire  wrapped around an iron core. When an electric current is applied, a magnetic field is created around the iron core. The magnetic field disappears when the power supply is interrupted. The core of a wire-wound magnet is ferromagnetic or ferromagnetic in nature. Iron is the most commonly used material in the core.

The formed magnetic field is controlled by electric power. A permanent magnet does not require a power source and the magnetic field cannot be controlled. Electromagnets are used to make solenoids, MRI machines, hard drives, relays, motors, speakers and generators.

A solenoid is a type of electromagnet designed to create a controlled magnetic field through a coil wound in a tightly packed coil. The solenoid is as shown below. The solenoid is a coil of wire and the piston is made of soft iron. A magnetic field is created around the coil when an electric current passes through it and pulls the piston. A solenoid can be said to be responsible for converting electrical energy into mechanical work.

Solenoids were born from the development of efficient and stronger magnets. In 1823, the French physicist and mathematician Andre-Marie Ampere coined the term solenoid to indicate a spiral coil.

Note. A solid core electromagnet is not considered a solenoid.

Operation of solenoids

After understanding the definition of a solenoid and its parts, let's learn how solenoids work. The solenoid works on the principle of electromagnetism. When an electric current passes through the coil, a magnetic field is created. When a metal core is placed inside the coil, the magnetic flux lines converge in the core. This increases the inductance of the coil compared to the air core. This concept is called electromagnetic induction.

As we know coils are made of many turns of copper wire tightly wound around it. A strong magnetic flux occurs when current flows through a conductor. The flux concentration is high in the core, while some flux is seen at the winding ends and also outside the winding. By increasing the current flow or increasing the rotation frequency, the magnetic strength of the solenoid can be increased.

The body is made of iron or steel. The magnetic field created by the coil surrounds the case. The piston provides the mechanical force for operation. Solenoids also have positive and negative terminals through which an object can be attracted or repelled.

Consider a solenoid whose length is greater than its radius. Enameled and insulated wire is wound with wire in the form of a thread with a small gap between the turns. The magnitude of the vector force produced by each turn and the total field produced by the solenoid is equal to the field produced by the circular loop. This provides the total field of the solenoid. The magnetic field lines produced inside a confined solenoid are shown below. The picture below shows a solenoid.

The field behind the solenoid is very thin. The field inside the solenoid is always parallel to its axis. The magnetic force produced by a solenoid can be determined using Ampere's law:

Where n is the number of turns of the wire per unit length, I is the current through the wire, and the direction is given by the right-hand rule.

Types of solenoid valves

The different types of solenoid valves are:

•  AC coated magnet
•  DC-C frame magnet
•  DC-D frame magnet
•  Linear solenoid
•  rotating magnet

AC coated magnet

AC stack solenoids are known for the amount of force they can generate on the first hit. AC stack solenoid valves have a metal core and a metal coil. The solenoid core is made of shielded metal to reduce stray current. Shielded AC solenoids are capable of more travel than shielded DC solenoids. This type of solenoid valve emits a clear hissing sound when turned on.

C frame DC solenoid valve

C-frame DC solenoid valves have a C-frame wrapped around the coil. C-frame DC solenoids are said to be in a DC configuration and can be used in devices designed for AC power.

DC-D frame Solenoid

It is a two-piece frame that covers the coil. The lifting movement is controlled and can also work with alternating current solenoid.

linear Solenoid

This linear Solenoid features a coil of wire wrapped around a moving metal core. Useful for applying tensile or compressive forces to mechanical equipment. A linear Solenoid is used to start the device. This mechanism helps close the circuit and allow electricity to pass through solenoid.

Rotating Solenoid:

A rotating magnet is used for the automatic control process. This magnet also has a coil and a core, but it works differently.

Applications of solenoid valves

It is used for construction vehicles, agricultural vehicles, special vehicles, industrial machinery, etc. Linear magnets are used in high security door mechanisms

Used in bike and car starters. Used in slot machines, scanners, printers, circuit breakers, photo shutters, etc.

Toroid:

A hollow circular ring in the shape of a donut. It consists of many turns of enameled wire, with almost no gap between the two turns.

A toroid can be thought of as a circular magnet used in a circuit as an inductor at low frequencies when large inductance is required.

The first toroidal core was invented by physicist Michael Faraday in his 1830s. He noticed that changing the magnetic field created tension in the wire. This phenomenon is known as Faraday's induction.

The magnetic field of a toroidal core is calculated by applying the Ampere Circuit Law :

Imagine a live wire wrapped many times around a hollow circular ring. In the figure above, suppose there is a magnetic field B  at a point P  inside the toroid. In the figure above, the loops are shown as Ampere loops forming circles through point P, resulting in concentric circles within the toroid.

Due to the symmetric field, the magnitude is the same at all points on the circle and the fields are tangent. therefore,

If the loop has N turns, the net current through that area is NI.

where I is the current in the toroid.

Using Ampere's law,

From this, we can conclude that the magnetic field B is varying and not uniform across the cross-section.

Toroid formula:

The toroidal core formula is used to calculate the number of turns in a toroidal coil.

Above is a pictorial representation of the cross-section of the inner diameter of the toroid and wire. let me

A is the inner radius of the toroid