Question

Two identical coaxial coils of wire of radius \(20.0 \mathrm{~cm}\) are directly on top of each other, separated by a 2.00 -mm gap. The lower coil is on a flat table and has a current \(i\) in the clockwise direction; the upper coil carries an identical current and has a mass of \(0.0500 \mathrm{~kg} .\) Determine the magnitude and the direction that the current in the upper coil has to have to keep the coil levitated at its current height.

Short answer

Answer: To levitate the upper coil at its current height, the magnitude and direction of the current required can be determined by following the steps outlined in the solution: applying Ampere's law to find the magnetic field of each coil, finding the force created by the interaction of these fields, and setting this force equal to the force of gravity. The direction of the current in the upper coil should be opposite to the current in the lower coil in order to counteract gravity and achieve levitation.

Step-by-step solution

Determine the magnetic field produced by each coil

To determine the magnetic field produced by each coil, we need to apply Ampere's law, which gives us the magnetic field around a closed loop as a function of the current enclosed by the loop. For a single turn coil, Ampere's law is given by \(B = \frac{\mu_0 I}{2R}\), where \(B\) is the magnetic field, \(\mu_0= 4\pi \times 10^{-7} T\cdot m/A\) is the permeability of free space, \(I\) is the current, and \(R= 20.0\text{ cm}\) is the radius of the coil.

Calculate the force created by the interaction of the magnetic fields

To calculate the force created by the interaction of the magnetic fields, we need to find the difference in the magnetic fields produced by the lower coil and upper coil. The force per unit length acting on the upper coil is given by \(F_{per\_length}=\frac{\mu_0 I_{lower} I_{upper}}{2\pi R_{gap}}.\) Here, \(R_{gap} = 2\text{ mm}\) is the distance between the two coils. Multiply this result by the circumference of the coil to get the total force acting on the upper coil, \(F_{total}= 2\pi R F_{per\_length}.\)

Determine the force of gravity acting on the upper coil

To find the gravitational force acting on the upper coil, we will use the equation \(F_{gravity} = mg,\) where \(m=0.0500\text{ kg}\) is the mass of the upper coil and \(g= 9.80\text{ m/s^2}\) is the acceleration due to gravity.

Set the force due to magnetic interaction equal to the gravitational force

For the upper coil to levitate at its current height, the force due to the magnetic interaction should be equal to the gravitational force acting on it. Set the total force equal to the gravitational force and solve for the current in the upper coil: \(2\pi R \frac{\mu_0 I_{lower} I_{upper}}{2\pi R_{gap}} = mg.\) Solve for \(I_{upper}\): \(I_{upper}= \frac{mgR_{gap}}{\mu_0 I_{lower}R}.\) We know the value of \(m, g, R_{gap}, \mu_0, I_{lower}\), and \(R\). Using this information, we can find the magnitude of the current in the upper coil.

Determine the direction of the current in the upper coil for levitation

Now that we have found the magnitude of the current required to levitate the upper coil, we need to determine the direction. For the force to be upwards and counteract gravity, the direction of the current in the upper coil has to be opposite to the current in the lower coil since they are directly on top of each other. If the lower coil's current is clockwise, the upper coil's current should be counterclockwise for levitation. By following these steps, we can determine both the magnitude and direction of the current in the upper coil required to keep the coil levitated at its current height.

Key concepts

Ampere's Law and its Role in Electromagnetic Levitation

Ampere's law is a fundamental principle in electromagnetism stating that for any closed loop, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability of free space times the electric current enclosed by the loop. In the context of electromagnetic levitation, this law allows us to calculate the magnetic field produced by an electric current flowing through a coil.

Using Ampere's Law, \(B = \frac{\mu_0 I}{2R}\), we can establish the relationship between the current \(I\) in a coil of radius \(R\), and the magnetic field \(B\) it generates. This magnetic field is what interacts with other currents or magnetic materials to produce a force, and in the case of our textbook exercise, it is key to levitating a coaxial coil above another.

Magnetic Field: The Invisible Force Behind Levitation

The magnetic field is an essential concept when discussing electromagnetic levitation. It's the area surrounding a magnet or a current-carrying coil where magnetic forces can be detected. Magnetic fields exert forces on other moving charges or magnetic materials within these fields.

In our levitation problem, the interaction between two magnetic fields created by the coils is utilized. The magnetic field originating from the lower coil induces a force on the upper coil, which can oppose the force of gravity when the currents are set up correctly. Strategically generating and manipulating these invisible magnetic fields provides the remarkable ability to levitate objects without any physical contact.

Gravitational Force: The Natural Phenomenon Counteracted by Levitation

Gravitational force is the natural phenomenon that attracts two bodies with mass. According to Newton's law of universal gravitation, this force is dependent on the mass of the objects and the distance between them. For objects on or near the surface of the Earth, gravitational force causes them to have weight and can be calculated using \(F_{gravity} = mg\), where \(m\) is the mass and \(g\) is the acceleration due to gravity.

In electromagnetic levitation, the force produced by the interaction of magnetic fields must be strong enough to counteract the gravitational force, effectively making the object levitate. The levitating object, our upper coil with mass \(0.0500 \text{kg}\), experiences a gravitational pull downward, which must be precisely balanced by the magnetic force pointing upward for stable levitation.

Coaxial Coils: Harnessing Magnetic Fields for Levitation

Coaxial coils are two or more coils that share a common axis. When currents flow through these coils, they produce magnetic fields. If arranged properly, as in our exercise, one coil can levitate over the other. The arrangement and interaction of coaxial coils are crucial for applications such as magnetic levitation trains, known as Maglev, and even in experimental setups to demonstrate scientific principles such as electromagnetic induction and levitation.

The distance between the coils and the orientation of the currents flowing through them determine the nature of the magnetic interaction. As seen in our textbook problem, a slight 2 mm gap separates the two coils, and the direction of current flow is critical for the levitation to occur. The forces generated by these currents in coaxial coils offer a vivid example of electromagnetism in action.