Exercises: Electric and magnetic field lines

State which of the following statements about electric field lines are true and which are false, correcting the false ones:

1. a) They start on positive charges and end on negative charges.
2. b) They can cross each other if the charges are sufficiently large.
3. c) Where they are closer together, the field is stronger.
4. d) The electric field vector at each point is tangent to the field line passing through it.

Magnetic field lines, unlike electric field lines, are always closed loops.

1. a) Describe the path they follow in a bar magnet: where they emerge, where they enter, and what they do inside the magnet.
2. b) Using the above, explain why breaking a magnet in half does not yield an isolated north pole and south pole, but rather two complete magnets.

Summarise in two or three sentences the fundamental difference between electric field lines and magnetic field lines in terms of where they begin and end, and relate it to these two facts: isolated electric charges (positive or negative) do exist, and isolated magnetic poles have not been observed.

1. a) Draw in your notebook the field lines of a positive point charge and, separately, those of a negative point charge, indicating with arrows the direction in each case.
2. b) Explain how the spacing between the lines changes as you move away from the charge and what this indicates about the field strength.

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

1. a) Draw in your notebook the field lines of a bar magnet, showing the north and south poles and the direction of the lines, both outside and inside the magnet.
2. b) State in which areas of the drawing the lines are closest together and explain what this means regarding the field at the poles.

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

An upward current flows through a straight vertical wire.

1. a) Draw in your notebook, in a horizontal plane perpendicular to the wire, the lines of its magnetic field: indicate their shape, their direction (right-hand rule), and how their spacing changes with distance from the wire.
2. b) Relate this increasing spacing to the dependency $B = \frac{\mu_0\times I}{2\pi\times r}$.

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

On a magnetic field line map, the lines appear very close together in one area and far apart in another. Look at the map:

1. a) Where is the field most intense? Justify your answer.
2. b) If a small compass is placed at any point on the map, how will it align itself relative to the line passing through that point?

In Oersted's experiment (1820) it was observed for the first time that an electric current deflects a nearby compass needle. The setup is shown in the figure (top view).

1. a) What did this observation demonstrate about the relationship between electricity and magnetism?
2. b) The needle tends to align perpendicular to the wire: what shape must the lines of the magnetic field created by the current have, then? Justify your answer.

Two identical positive charges $+q$ are close to each other.

1. a) Qualitatively draw in your notebook the field line map of the system.
2. b) Locate the point where the total electric field is zero and justify it by symmetry.
3. c) Reason why no line leaving one of the charges ends on the other.

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

1. a) Draw in your notebook the field line map of an electric dipole (a $+q$ charge and a $-q$ charge close together).
2. b) Separately, draw the field line map of a bar magnet.
3. c) Compare both maps: how are they similar on the outside? How do they fundamentally differ, considering whether the lines form closed loops or not and what happens in the region between the poles?

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

The magnetic field lines of a straight wire are concentric circles that get further apart as they move away from the wire.

1. a) Draw the situation in your notebook, with the current coming out of the page.
2. b) Using $B = \frac{\mu_0\times I}{2\pi\times r}$, calculate the ratios $\frac{B(r)}{B(2\times r)}$ and $\frac{B(r)}{B(3\times r)}$ and explain why the lines should be drawn further and further apart.
3. c) How should a uniform magnetic field be drawn using this same line convention?

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

1. a) Draw in your notebook the field line map of a current-carrying solenoid, both inside and outside.
2. b) Explain, based on your drawing, why the inner field is much more intense and almost uniform, whereas the outer field is weak.
3. c) Compare the outer map with that of a bar magnet: what similarity do you observe?

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

Below are three separate maps of magnetic field lines: that of a current-carrying loop (seen edge-on), that of a short solenoid and that of a bar magnet. For simplicity, the arrows showing the direction of the lines have been omitted from the maps, and the direction of the current is not shown.

1. a) Explain how the three maps are similar.

1. b) Why is a current-carrying loop said to behave like a small magnet?

1. c) If the current in the loop is counterclockwise when viewed from one of its faces, does that face act as a north pole or as a south pole? Justify your answer using the right-hand rule.

The Earth's magnetic field resembles that of a giant bar magnet, slightly tilted relative to the Earth's axis of rotation.

1. a) Draw in your notebook, qualitatively, the lines of the Earth's magnetic field around the planet.
2. b) A compass points with its north towards geographic north: reason why this means that near geographic north there is actually a magnetic south pole.
3. c) Bearing in mind that lines enter through the magnetic south pole, do they enter or leave the ground in the Northern Hemisphere? What components does the Earth's field have there: horizontal, vertical, or both?

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

When drawing field maps, the number of lines starting or ending at a charge is taken to be proportional to its value. Two point charges, $+2\times q$ and $-q$, are close together.

1. a) How many lines leave the positive charge for every line that ends on the negative charge?
2. b) If all the lines ending on the negative charge come from the positive charge, what happens to the "surplus" lines from the positive charge?
3. c) Qualitatively draw the map in your notebook.

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

A flat loop is placed in a region with a uniform magnetic field.

1. a) In which orientation of the loop does the greatest possible number of field lines pass through it? Draw it in your notebook.
2. b) And in which orientation does no field line pass through it?
3. c) As the loop is slowly rotated from one orientation to the other, how does the number of lines passing through it change? This idea is the basis for the concept of magnetic flux, $\Phi$, which you will study in the induction topic.

Prove by proof by contradiction that two field lines (electric or magnetic) can never cross. The assumed situation is shown in the figure:

1. a) Suppose two lines intersect at a point and draw in your notebook the tangents to both at that point.
2. b) Recall what relationship the tangent to a line has with the field vector at each point: how many distinct field vectors would there be then at the intersection point?
3. c) Conclude why this situation is impossible and state the resulting property.

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

Two identical bar magnets are placed on a table with their north poles facing each other. Look at the setup:

1. a) Qualitatively draw in your notebook the line map in the region between both magnets.
2. b) Locate the neutral point (point of zero field) and justify it by superposition of the fields from each magnet.
3. c) Repeat the analysis when a north pole and a south pole face each other: is there a neutral point between them now? What are the lines like in the intermediate region and what does this imply about the force between the magnets in each case?

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

Two straight parallel wires, separated by $d = 10\ \text{cm}$, carry equal currents $I_1 = I_2 = 15\ \text{A}$ in the same direction ($\mu_0 = 4\pi\times10^{-7}\ \text{T}\times\text{m/A}$).

1. a) Draw in your notebook, in the plane perpendicular to the wires, the field lines of each wire separately and, superimposed, the qualitative appearance of the total field.
2. b) Locate the point where the total field is zero and justify it.
3. c) At points at a distance $r$ much greater than $d$, justify that the map resembles that of a single wire carrying current $2\times I$ and calculate the field at $r = 1\ \text{m}$ from the pair of wires.

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

We repeat Oersted's experiment with measurements: a compass is placed $r = 2\ \text{cm}$ from a vertical wire and, when the circuit is closed, the needle rotates through an angle $\theta$ such that $\tan\theta = \frac{B_{cable}}{B_T}$, where $B_T = 5\times10^{-5}\ \text{T}$ is the horizontal component of the Earth's field ($\mu_0 = 4\pi\times10^{-7}\ \text{T}\times\text{m/A}$). Look at the diagram of the two perpendicular fields:

1. a) Calculate the field of the wire for $I = 5\ \text{A}$ and verify that the deflection is $45^\circ$.
2. b) Calculate what current is needed for the deflection to be $60^\circ$ ($\tan 60^\circ \approx 1.73$).
3. c) Explain, with the help of a drawing in your notebook of the two perpendicular fields, why the needle aligns along their resultant.

A Faraday cage (a cavity surrounded by conducting material) cancels out the external electric field inside it, because the charges in the conductor redistribute until they compensate for it.

1. a) Explain, using this mechanism, why the cage works with static electric fields.
2. b) Reason, from the fact that magnetic lines are closed loops and there are no "magnetic charges" that can redistribute, why a conducting cage does not shield a static magnetic field.
3. c) Real magnetic shielding uses high-permeability ferromagnetic materials, which "channel" the field lines through their interior. Relate this to the fact that field lines cannot simply terminate.

In a cloud chamber, a particle detector, the trajectories of charged particles inside a uniform magnetic field perpendicular to the image are photographed. Look at the image from the chamber:

1. a) What would the line map of this uniform field look like? Draw it in your notebook.
2. b) Several particles of the same charge describe circles of different radii: according to $r = \frac{m\times v}{q\times B}$, what is the difference between particles with a large radius and those with a small radius?
3. c) In reality, the trajectories are spirals that gradually narrow: reason what is happening to the speed and energy of the particles, recalling that the magnetic force does no work.

An ideal parallel-plate capacitor and an ideal long solenoid both generate a practically uniform field in their interior and an almost zero field outside.

1. a) Draw in your notebook, side by side, the field line maps of both devices.
2. b) Point out the similarities between both maps: interior uniformity, negligible outer field.
3. c) Point out the fundamental differences: what are the "sources" of the field in each case? Are the lines open or closed? What happens at the edges in real devices (edge effects)?

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

In a certain region the magnetic field is not uniform: at point A the field lines are much closer together than at point B, and measurements give $B_A = 0.4\ \text{T}$ and $B_B = 0.1\ \text{T}$. A proton ($m_p = 1.67\times10^{-27}\ \text{kg}$, $e = 1.6\times10^{-19}\ \text{C}$) enters with the same speed, $v = 2\times10^{6}\ \text{m/s}$, perpendicular to the field, first at A and then at B. The line map of the region is shown in the figure.

1. a) Calculate the radius of the trajectory at each point, $r = \frac{m\times v}{q\times B}$.
2. b) Calculate the ratio $\frac{r_B}{r_A}$ and relate it to the line density ratio $\frac{B_A}{B_B}$.
3. c) Conclude: on a field line map, where does the trajectory of a charge curve more, where the lines are close together or where they are far apart? Justify your answer.