Question

A steel cylinder with radius \(2.5 \mathrm{~cm}\) and length \(10.0 \mathrm{~cm}\) rolls without slipping down a ramp that is inclined at \(15^{\circ}\) above the horizontal and has a length (along the ramp) of \(3.0 \mathrm{~m} .\) What is the induced potential difference between the ends of the cylinder at the bottom of the ramp, if the surface of the ramp points along magnetic north?

Short answer

Answer: To find the induced potential difference, we first determined the final velocity of the cylinder at the bottom of the ramp using conservation of energy principles. Next, we calculated the change in magnetic flux resulting from the rolling motion of the cylinder. Finally, we applied Faraday's law to find the induced potential difference between the ends of the cylinder at the bottom of the ramp.

Step-by-step solution

Determine the final velocity of the cylinder

To find the final velocity of the cylinder, we'll first calculate the gravitational potential energy of the cylinder at the top of the ramp and then equate it to the kinetic energy at the bottom of the ramp. Using the height \(h\) of the ramp and the mass \(m\) of the cylinder, the potential energy is given by \(E_\text{potential} = mgh\), where \(g = 9.81 \mathrm{~m/s^2}\) is the acceleration due to gravity. To find the height \(h\), we use the relation: $$ h = \sin(15°) \cdot 3.0 \mathrm{~m} $$ The kinetic energy at the bottom of the ramp is given by \(E_\text{kinetic} = \frac{1}{2}mv^2\), where \(v\) is the final velocity. Equating potential and kinetic energies, we have: $$ mgh = \frac{1}{2}mv^2 $$ Solving for \(v\), we get: $$ v = \sqrt{2gh} $$

Calculate the change in magnetic flux

To determine the change in magnetic flux, we'll first find the flux through the top and bottom surfaces of the cylinder. The magnetic field at the Earth's surface is approximately \(B = 50 \times 10^{-6} \mathrm{T}\). The area of the top and bottom surfaces of the cylinder is given by \(A = \pi r^2 = \pi (0.025 \mathrm{m})^2\), where \(0.025\,\mathrm{m}\) is the radius of the cylinder in meters. Therefore, the magnetic flux through both top and bottom surfaces is given by: $$ \Phi_{\text{top}} = \Phi_{\text{bottom}} = B \cdot A $$ Since the cylinder rolls without slipping, the top and bottom surfaces change their orientation throughout its motion, such that the top surface becomes vertical (parallel to the magnetic field) at the bottom of the ramp. Therefore, at this point, the magnetic flux through the top surface will be zero. So, the change in the magnetic flux will be: $$ \Delta \Phi = \Phi_{\text{bottom}} - 0 $$

Find the induced potential difference

Finally, we need to apply Faraday's law of electromagnetic induction, which states that the induced electromotive force (EMF) is equal to the negative rate of change of the magnetic flux. The induced potential difference \(V\) is equal to the induced EMF. Considering that the rotation of the cylinder will take time \(t = \frac{3.0\mathrm{~m}}{v}\), we can write Faraday's law as: $$ V = -\frac{d(\Delta \Phi)}{dt} = -(B \cdot A) \cdot \frac{1}{t} $$ Plugging in the previously calculated values for the final velocity, magnetic field, and area, we can obtain the induced potential difference \(V\) between the ends of the cylinder at the bottom of the ramp.

Key concepts

Faraday's Law

Faraday's Law is a fundamental principle that describes how electric currents are induced by changes in magnetic flux. Essentially, it states that an electromotive force (EMF) is generated in a closed circuit when the magnetic flux through that circuit changes. To visualize this, imagine a loop of wire exposed to a magnetic field; as the field fluctuates, electricity flows through the loop.

Mathematically, Faraday's Law is represented as \( E = -\frac{d\Phi}{dt} \), where \( E \) is the induced EMF, \( \Phi \) represents magnetic flux, and the negative sign indicates the direction of the induced EMF opposes the change in flux (as described by Lenz's Law). To apply this to the steel cylinder example, the final velocity is used to determine the time it would take for the cylinder to roll down the ramp. This, in turn, helps to calculate the rate of change of magnetic flux which allows us to find the induced EMF or potential difference across the cylinder.

Electromagnetic Induction

Electromagnetic induction is the process by which a conductor moving through a magnetic field produces an electrical current. This remarkable phenomenon is the working principle behind generators, transformers, and many types of sensors. It is deeply linked with Faraday's Law of induction, which provides the quantitative measurement of this induced current or EMF.

In the cylinder problem, electromagnetic induction occurs because as the cylinder rolls down the ramp, different parts of it move through the Earth's magnetic field. This movement alters the magnetic flux through the circular cross-section of the cylinder, and as a result, an EMF is induced between the ends of the cylinder. It is essential to note that for induction to occur, the conductor must experience a change in magnetic flux, which in this case is caused by the rotation and translation of the cylinder.

Kinetic and Potential Energy

Kinetic and potential energy are two fundamental forms of mechanical energy. Potential energy is stored energy due to an object's position or configuration, while kinetic energy is the energy of motion. An object at a height has gravitational potential energy given by \( E_{\text{potential}} = mgh \), with \( m \) being mass, \( g \) gravitational acceleration, and \( h \) the height above a reference point.

In the context of our steel cylinder, it starts with potential energy at the top of the ramp, which is gradually converted to kinetic energy as it rolls down. When the cylinder reaches the bottom, its potential energy has been fully transformed into kinetic energy, resulting in the final velocity. This transformation is utilized to calculate the potential difference induced in the cylinder, as the conversion rate from potential to kinetic energy (reflected in the final velocity) provides insights into the timing of the electromagnetic induction event.