Generating the Electromotive Force

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.

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.