Generating the Electromotive Force

A square loop of side $20\ \text{cm}$ is placed in a uniform magnetic field of $0.5\ \text{T}$, with the field perpendicular to the plane of the loop, as shown in the figure.

1. a) Calculate the area of the loop in $\text{m}^2$.
2. b) Calculate the magnetic flux passing through it, $\Phi = B \times S \times \cos \alpha$, stating the value of the angle $\alpha$ that the field makes with the normal to the surface.

In an MRI room, a flat search coil of $10\ \text{cm}^2$ is oriented so that its normal makes an angle of $30^\circ$ with a uniform magnetic field of $0.8\ \text{T}$ (take $\cos 30^\circ \approx 0.87$). Look at the diagram:

1. a) Explain what the angle $\alpha$ in the expression $\Phi = B \times S \times \cos \alpha$ represents.
2. b) Calculate the magnetic flux passing through the coil.

The magnetic flux through a loop of area $0.05\ \text{m}^2$, placed perpendicular to a uniform field, is $0.1\ \text{Wb}$.

1. a) Calculate the magnitude of the magnetic field.
2. b) Express the weber in terms of the tesla and the square metre.

A square loop of side $12\ \text{cm}$ rotates inside a field of $0.25\ \text{T}$ and moves from the position perpendicular to the field (maximum flux) to the position parallel to the field (zero flux) in $0.5\ \text{s}$.

1. a) Calculate the maximum flux passing through the loop.
2. b) Calculate the change in flux during this rotation.
3. c) Calculate the average induced emf.

When bringing a contactless card close to the reader, the flux passing through the loop in its chip changes from $0$ to $4\times10^{-4}\ \text{Wb}$ in $0.01\ \text{s}$.

1. a) Calculate the average electromotive force induced in the loop, $\varepsilon = \frac{\Delta \Phi}{\Delta t}$.
2. b) Explain, using Faraday's law, why no emf is induced if the card remains stationary next to the reader, assuming that the reader's magnetic field is constant in time.

The flux passing through each turn of a coil with $200$ turns changes from $5\times10^{-3}\ \text{Wb}$ to $2\times10^{-3}\ \text{Wb}$ in $0.1\ \text{s}$.

1. a) Calculate the change in flux, $\Delta \Phi$.
2. b) Calculate the induced emf in the coil, taking into account that it must be multiplied by the number of turns: $\varepsilon = -N \times \frac{\Delta \Phi}{\Delta t}$.

A circular loop of radius $4\ \text{cm}$ is placed perpendicular to a magnetic field that increases uniformly from $0.2\ \text{T}$ to $0.8\ \text{T}$ in $0.05\ \text{s}$ (take $\pi \approx 3.14$).

1. a) Calculate the area of the loop in $\text{m}^2$.
2. b) Calculate the change in flux passing through it.
3. c) Calculate the average induced emf.

A coil with $100$ turns has a total resistance of $5\ \Omega$ and each turn has an area of $50\ \text{cm}^2$. The field perpendicular to its plane decreases uniformly from $0.5\ \text{T}$ to $0.1\ \text{T}$ in $0.2\ \text{s}$.

1. a) Calculate the change in flux in each turn.
2. b) Calculate the induced emf in the coil.
3. c) Calculate the current flowing through it by applying Ohm's law.

In a demonstration setup, two coils share a soft iron core, as shown in the figure. When the switch in the first circuit is closed, the field in the core changes from $0$ to $0.06\ \text{T}$ in $0.02\ \text{s}$. The second coil has $300$ turns of $10\ \text{cm}^2$ each.

1. a) Calculate the change in flux in each turn of the second coil.
2. b) Calculate the induced emf in the second coil.
3. c) Explain why the emf disappears a moment later, when the current in the first circuit becomes constant.

The north pole of a magnet is brought close to a circular copper loop, viewing the loop from the magnet's side, as shown in the figure.

1. a) According to Lenz's law, which magnetic pole appears on the face of the loop facing the magnet? Justify your answer.
2. b) In which direction does the induced current flow when viewed from the magnet, clockwise or anticlockwise? Recall that a face around which current flows anticlockwise behaves as a north pole.

State, justifying your answer in each case, in which of these situations an electromotive force is induced in the coil:

• A magnet at rest inside a coil.

• A magnet entering the coil.

• A coil carrying a constant direct current, placed next to another coil.

• The same pair of coils as in the previous case, at the moment the switch is opened and the current turns off.

The south pole of a magnet is moved away from a coil connected to a galvanometer, as shown in the figure.

1. a) What magnetic pole appears on the face of the coil near the magnet as it moves away? Reason your answer using Lenz's law.
2. b) If the magnet were to move more slowly, would the galvanometer measure more or less current? Which law explains this?

A circular elastic loop is in a magnetic field of $0.3\ \text{T}$ perpendicular to its plane. It is stretched so that its area changes from $100\ \text{cm}^2$ to $200\ \text{cm}^2$ in $0.1\ \text{s}$.

1. a) Calculate the change in flux.
2. b) Calculate the induced emf.
3. c) Reason, using Lenz's law, whether the induced current creates a magnetic field in the same direction as the external field or in the opposite direction.

The flat coil of a sensor has $50$ turns of $20\ \text{cm}^2$ each and is in a field of $0.1\ \text{T}$. On rotating the coil through a quarter-turn in $0.01\ \text{s}$, the flux goes from its maximum value to zero, as shown in the figure.

1. a) Calculate the maximum flux passing through each turn.
2. b) Calculate the average induced emf in the entire coil.

Two identical magnets are dropped through two identical vertical tubes, one made of copper and the other of plastic, as shown in the figure. It is observed that the magnet falls much more slowly through the copper tube.

1. a) Explain, using Faraday's and Lenz's laws, why induced currents appear in the copper tube and why they slow down the magnet.
2. b) Where does the energy dissipated as heat in the copper come from? Analyse the conservation of energy in the experiment.
3. c) Predict what would happen if the copper tube had a longitudinal slot along its entire length, and give reasons for your answer.

The magnetic field perpendicular to the plane of a coil of $100$ turns of $30\ \text{cm}^2$ varies as follows: during the first $2$ seconds it increases uniformly from $0$ to $0.6\ \text{T}$; during the following $2$ seconds it remains constant; and between $t = 4\ \text{s}$ and $t = 6\ \text{s}$ it decreases uniformly to $-0.6\ \text{T}$, reversing its direction.

1. a) Calculate $\frac{\Delta B}{\Delta t}$ in each of the three sections.
2. b) Calculate the induced emf in each section.
3. c) Applying Lenz's law, compare the direction of the current in the first section with that in the third (note: in the third section, the field first decreases to zero and then grows in the opposite direction; analyse whether the rate of change of flux changes or not within that section).
4. d) Draw the graphs of $B(t)$ and $\varepsilon(t)$ one below the other in your notebook, with the axes aligned.

You can do it on paper and send a photo with your answer.

A laboratory trolley carries a rectangular loop and advances at $2\ \text{m/s}$ entering the region between the poles of an electromagnet, where the field is $0.4\ \text{T}$ and is perpendicular to the plane of the loop. The side of the loop transverse to the motion measures $15\ \text{cm}$. The situation is represented in the figure.

1. a) Calculate the induced emf while the loop is entering, using $\varepsilon = B \times l \times v$.
2. b) Explain why the emf becomes zero when the loop is completely inside the field.

A low-orbit satellite deploys a conducting cable $2\ \text{km}$ long and moves at $7.5\times10^{3}\ \text{m/s}$ in a direction perpendicular to the Earth's magnetic field, which at that altitude is $3\times10^{-5}\ \text{T}$. The situation is represented in the figure.

1. a) Calculate the potential difference that appears across the ends of the cable, $\varepsilon = B \times l \times v$.
2. b) Explain why an emf is only induced when the cable moves cutting the field lines, and not if it moves in the direction of the field.

A square loop with $10\ \text{cm}$ sides and electrical resistance $R = 0.5\ \Omega$ moves at a constant speed of $1\ \text{m/s}$ and passes through a square region with $20\ \text{cm}$ sides where there is a uniform magnetic field of $0.5\ \text{T}$, perpendicular to the plane of the loop and pointing inward (into the paper), as shown in the figure. The loop moves to the right.

1. a) Calculate the emf while the loop is entering the region ($\varepsilon = B \times l \times v$) and the time the entry lasts.
2. b) What is the emf while the loop is completely inside? Why?
3. c) What emf is there while it exits, and what is the direction of the current compared with that during the entry? Apply Lenz's law.
4. d) Draw in your notebook the graph of the current as a function of time over the whole crossing (from when the loop starts to enter until it finishes exiting), indicating the numerical values and the signs. Use the resistance of the loop, $R = 0.5\ \Omega$, and take the direction of the current during the entry as positive.

You can do it on paper and send a photo with your answer.

To extract a rectangular loop at a constant speed from a field of $0.8\ \text{T}$, it must be pulled with a force, as shown in the figure. The loop has a resistance of $0.1\ \Omega$, its side transverse to the motion measures $20\ \text{cm}$ and it is extracted at $0.5\ \text{m/s}$.

1. a) Calculate the induced emf during extraction ($\varepsilon = B \times l \times v$).
2. b) Calculate the induced current.
3. c) Calculate the magnetic force on the transverse side, $F = I \times l \times B$, and explain why, according to Lenz's law, it opposes the extraction.
4. d) Calculate the mechanical power of the external force, $P = F \times v$, and the electrical power dissipated in the loop, $P = \varepsilon \times I$; compare them and explain what physical principle this agreement illustrates.

An aeroplane flies horizontally northwards at $900\ \text{km/h}$ with a wingspan of $35\ \text{m}$. In that area, the vertical component of the Earth's magnetic field is $4\times10^{-5}\ \text{T}$ and points towards the ground. Observe the diagram (top view):

1. a) Express the speed in $\text{m/s}$.
2. b) Calculate the potential difference that appears between the wingtips, $\varepsilon = B \times l \times v$.
3. c) The positive charges in the metal experience the magnetic force from Lorentz's law: applying the right-hand rule to the vector product of $\vec{v}$ by $\vec{B}$, reason which wingtip, east or west, is at a higher potential.

A coil with $500$ turns of $40\ \text{cm}^2$ each and total resistance $20\ \Omega$ is in a magnetic field perpendicular to its plane that varies, in SI units, according to the law $B = 0.8 - 2 \times t$. The situation is represented in the figure.

1. a) Calculate the rate of change of the field, $\frac{\Delta B}{\Delta t}$.
2. b) Calculate the induced emf in the coil and the current flowing through it.
3. c) Calculate the charge passing through a cross-section of the circuit in $0.3\ \text{s}$.
4. d) If the field points into the page and is decreasing, reason using Lenz's law whether the induced current is clockwise or anticlockwise as viewed from your side.

In an industrial induction furnace, a coil around the crucible generates an alternating field: in half a cycle, the field changes from $-0.05\ \text{T}$ to $+0.05\ \text{T}$ in $1\times10^{-5}\ \text{s}$. A section of the metal in the crucible is equivalent to a loop of $100\ \text{cm}^2$ perpendicular to the field.

1. a) Calculate the change in flux in half a cycle and the average induced emf in that equivalent loop.
2. b) Explain why the furnace requires alternating current and would not work with a constant direct current.
3. c) Explain why the metal in the crucible heats up due to the Joule effect, and explain what eddy currents are.

To measure the field between the poles of a horseshoe magnet, a search coil of $50$ turns of $4\ \text{cm}^2$ each is used, with a total resistance of $10\ \Omega$, connected to a charge meter. It is placed perpendicular to the field and abruptly removed in $0.2\ \text{s}$; the meter registers a charge of $4\times10^{-4}\ \text{C}$.

1. a) Chain the relationships $q = I \times \Delta t$, $I = \frac{\varepsilon}{R}$ and $\varepsilon = N \times \frac{\Delta \Phi}{\Delta t}$ to deduce that $B = \frac{q \times R}{N \times S}$.
2. b) Calculate the magnitude of the field between the poles of the magnet.
3. c) Calculate the average emf and the average current during extraction.