Irrigation channels that require regular flow monitoring are...

Irrigation channels that require regular flow monitoring are often equipped with electromagnetic flow meters in which the magnetic field is produced by horizontal coils embedded in the bottom of the channel. A particular coil has 100 turns and a diameter of $6.0 \mathrm{m}$. When it's time for a measurement, a $5.0 \mathrm{A}$ current is turned on. The large diameter of the coil means that the field in the water flowing directly above the center of the coil is approximately equal to the field in the center of the coil. a. What is the magnitude of the field at the center of the coil? b. If the field is directed downward and the water is flowing east, what is the direction of the force on a positive ion in the water above the center of the coil? c. If the water is flowing above the center of the coil at $1.5 \mathrm{m} / \mathrm{s}$, what is the magnitude of the force on an ion with a charge $+e ?$

Irrigation channels that require regular flow monitoring are often equipped with electromagnetic flow meters in which the magnetic field is produced by horizontal coils embedded in the bottom of the channel. A particular coil has 100 turns and a diameter of $6.0 \mathrm{m}$. When it's time for a measurement, a $5.0 \mathrm{A}$ current is turned on. The large diameter of the coil means that the field in the water flowing directly above the center of the coil is approximately equal to the field in the center of the coil.
a. What is the magnitude of the field at the center of the coil?
b. If the field is directed downward and the water is flowing east, what is the direction of the force on a positive ion in the water above the center of the coil?
c. If the water is flowing above the center of the coil at $1.5 \mathrm{m} / \mathrm{s}$, what is the magnitude of the force on an ion with a charge $+e ?$

Key Concepts

Right-Hand Rule in Electromagnetism

The right-hand rule is a mnemonic used to determine the direction of the force exerted on a moving charge in a magnetic field, based on the cross product in the Lorentz force law. By pointing the fingers in the direction of the velocity and curling them toward the magnetic field, the thumb then points in the direction of the force acting on a positive charge. This tool is essential for predicting the behavior of charged particles in electromagnetic applications.

Lorentz Force

The Lorentz force law describes the force experienced by a charged particle moving in a magnetic field. This force is given by the cross product of the particle's velocity and the magnetic field, scaled by the particle's charge. It forms the basis of the operating principle of electromagnetic flow meters, where moving charges in a conductive fluid experience forces that can be related to the flow velocity.

Magnetic Field of a Circular Coil

This concept involves calculating the magnetic field generated by a current-carrying coil. The field at the center of a circular coil can be derived using the Biot–Savart law or Ampere’s law, leading to an expression that depends on the number of turns, the current flowing through the coil, and the coil’s radius. A larger coil diameter influences the uniformity of the magnetic field in the region above the coil, making simplifications possible when evaluating the field at the center.

Transcript

00:01 High in the given problem the number of turns in the coil is given as n is equal to hundred diameter of this coil is given as 6 .0 meter so its radius r will be given as half of the diameter means 6 .0 meter by 2 means this is r is equal to 3 .0 meter, the current passing through this coil is i is equal to 5 .0 ampere.

00:49 Now in the first part of the problem, we have to find the magnetic field produced and the center of this coil.

00:57 So magnetic field produced at the center of the coil is given as b is equal to.

01:16 To mu not n i by 2 r.

01:23 Here for mu not this is 4 pi into 10 dash per minus 7 tesla meter per ampier into for the number of turns this is 100 for current this is 5 amper divided by 2 multiplied by radius which is 3 .0 meter.

01:45 So finally, this magnetic field here comes out to be 10 .5 into 10 dash per minus 5 tesla, which becomes the answer for the first part of this problem.

02:08 In the second part of the problem, the direction of flow of the water, here at the center of this coil, the direction of magnetic field, which we have just found, its direction here is given as vertically downward, and the water is flowing east.

02:38 So, if we consider the directions to be north, south, and east, west to be here, the direction of the direction of, the motion of the positive ions will be same as that of the direction of flow of the water.

03:02 Means this is v, the velocity vector.

03:06 Here this is b, the magnetic field vector.

03:10 And here, using the expression for the magnetic lawrence force experienced by the charge particles moving in magnetic field, that is given as f, is equal to q into v cross b...

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