Exercises: Electric potential energy and potential

A small sphere in the laboratory has a charge $Q = 3\,\mu\text{C}$ (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Calculate the electric potential at $0.5\ \text{m}$ from the sphere, $V = K \times \frac{Q}{r}$.
2. b) State the sign of the potential and reason where, by convention, $V = 0$ is taken.

Electric potential is a scalar quantity; electric field, a vector quantity.

1. a) Explain the practical difference this makes when adding up the contributions from several charges at a point.
2. b) Can the potential at a point be negative? Give an example.
3. c) What sign does the potential created by a positive charge have?

A charge $q = 2\,\mu\text{C}$ is located at a point where the potential is $V = 1000\ \text{V}$.

1. a) Calculate its electric potential energy, $E_p = q \times V$.
2. b) If the charge, which is positive, moved spontaneously towards a point of lower potential, reason whether its potential energy would increase or decrease.

From the expression for work $W = q \times \Delta V$:

1. a) Deduce the equivalence $1\ \text{V} = 1\ \text{J/C}$.
2. b) Explain what it means for there to be a potential difference of $12\ \text{V}$ between the terminals of a car battery, in terms of the energy of each coulomb of charge passing through it.
3. c) State which of these expressions are equivalent to one volt: $\text{J/C}$; $\text{N/C}$; $\text{J}\times\text{C}$.

A charge $q = 5\,\mu\text{C}$ moves from a point A, with potential $V_A = 200\ \text{V}$, to a point B, with $V_B = 80\ \text{V}$.

1. a) Calculate the work done by the electric field, $W_{A \to B} = q \times (V_A - V_B)$.
2. b) Reason, from the sign of the result, whether the field spontaneously "pushes" the charge in that direction of displacement.

In particle physics, the electronvolt is widely used: $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$ (given: elementary charge $e = 1.6\times10^{-19}\ \text{C}$).

1. a) Express an energy of $3\ \text{eV}$ in joules.
2. b) Justify the equivalence above by calculating, using $W = q \times \Delta V$, the energy gained by an electron when accelerated through a potential difference of $1\ \text{V}$.

Two point charges $q_1 = 1\,\mu\text{C}$ and $q_2 = 2\,\mu\text{C}$ are separated by $0.3\ \text{m}$ (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Calculate the electric potential energy of the system, $E_p = K \times \frac{q_1 \times q_2}{r}$.
2. b) Interpret the sign of the result: when separating the charges to infinity, must energy be supplied or is it released? Relate this to the fact that the charges repel each other.

There is a potential difference of about $70\ \text{mV}$ between the inside and outside of a cell, with the inside being negative (given: elementary charge $e = 1.6\times10^{-19}\ \text{C}$). The figure shows the diagram of the membrane.

1. a) Express this potential difference in volts and state how much energy is needed, at minimum, to transport one coulomb of positive charge from the inside to the outside against the field.
2. b) Calculate the energy required to transport a sodium ion $\text{Na}^+$ (charge $+e$) from the inside to the outside, in joules and in electronvolts.

In a laboratory Van de Graaff generator, a potential of $6000\ \text{V}$ is measured at $0.25\ \text{m}$ from the centre of the charged sphere (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$ and assume a point charge).

1. a) Rearrange for the charge in the expression $V = K \times \frac{Q}{r}$ and calculate it.
2. b) What would the potential be at twice the distance? Reason using the proportionality you have used.

Two charges $q_1 = 4\ \text{nC}$ and $q_2 = -2\ \text{nC}$ are at $0.3\ \text{m}$ and $0.5\ \text{m}$, respectively, from a point P (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Calculate the potential created by each charge at P.
2. b) Calculate the total potential at P.
3. c) Explain why here the contributions are added directly as numbers, whereas for the electric field vectors must be added.

A charge $q = -3\,\mu\text{C}$ moves from a point A ($V_A = 500\ \text{V}$) to a point B ($V_B = 100\ \text{V}$).

1. a) Calculate the work done by the field, $W_{A \to B} = q \times (V_A - V_B)$.
2. b) Interpret the negative sign of the result: is it a spontaneous movement or must it be carried out by an external agent?
3. c) Reason where this negative charge would move spontaneously: towards higher potentials or towards lower potentials?

A detector records high-energy particles (given: $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$).

1. a) A recorded particle has an energy of $2.4\times10^{-17}\ \text{J}$: express it in electronvolts.
2. b) An alpha particle emitted in a nuclear decay has an energy of $5\ \text{MeV}$: express it in joules.
3. c) Explain, by comparing the numbers above, why the electronvolt is a convenient unit in atomic and nuclear physics.

In an electron microscope, an electron starts from rest and is accelerated through a potential difference $\Delta V = 2000\ \text{V}$ (given: elementary charge $e = 1.6\times10^{-19}\ \text{C}$; electron mass $m_e = 9.11\times10^{-31}\ \text{kg}$). Observe the diagram:

1. a) Calculate the kinetic energy gained by the electron, in joules and in electronvolts.
2. b) Apply conservation of energy, $\frac{1}{2} \times m_e \times v^2 = e \times \Delta V$, and calculate its final speed.

For two protons to fuse inside the Sun, they must overcome their electric repulsion and get extremely close to each other (given: elementary charge $e = 1.6\times10^{-19}\ \text{C}$; $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$; $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$; $1\ \text{nm} = 10^{-9}\ \text{m}$). The figure shows the potential energy of the pair of protons as a function of their separation.

1. a) Calculate the electric potential energy of two protons separated by a distance $r = 1\ \text{nm}$.
2. b) Express the result in electronvolts.
3. c) Reason why electric repulsion hinders fusion and what role the extremely high temperature in the solar interior plays in overcoming it.

Two identical charges $q = 5\ \text{nC}$ occupy the points $(0, 0)$ and $(0, 40)$, with coordinates in centimetres (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Reason, drawing the vectors in your exercise book, that the total electric field at the midpoint $P(0, 20)$ is zero.
2. b) Calculate the electric potential at that same point P.
3. c) Draw a conclusion: is it true that where the field is zero, the potential is also zero?

Between the two flat electrodes of an ionisation smoke detector, separated by $d = 5\ \text{cm}$, there is a uniform electric field $E = 10^{4}\ \text{N/C}$ (given: elementary charge $e = 1.6\times10^{-19}\ \text{C}$). Observe the diagram:

1. a) Calculate the potential difference between the electrodes, $\Delta V = E \times d$.
2. b) Calculate the work done by the field on a positive ion of charge $+e$ moving from the positive electrode to the negative electrode.
3. c) Check that the result coincides with the value calculated as $W = F \times d = e \times E \times d$ and explain why this must be so.

In the acceleration stage of a proton therapy machine (treatment of tumours with proton beams), a proton starts from rest at a point A with $V_A = 3000\ \text{V}$ and reaches a point B with $V_B = 500\ \text{V}$ (given: elementary charge $e = 1.6\times10^{-19}\ \text{C}$; proton mass $m_p = 1.67\times10^{-27}\ \text{kg}$).

1. a) Calculate the potential energy lost by the proton between A and B.
2. b) Apply conservation of mechanical energy and calculate the speed of the proton at B.
3. c) Reason, using the sign of the charge and the change in potential energy, whether the proton could move spontaneously in the opposite direction, from B to A.

Two point charges $q_1 = 6\ \text{nC}$ and $q_2 = -2\ \text{nC}$ lie on the $x$ axis, at $x_1 = 0$ and $x_2 = 1\ \text{m}$ (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Set up the condition $V = 0$ at a point on the axis and express it as a relationship between the distances to each charge: what relationship must hold between $r_1$ and $r_2$?
2. b) Find the point on the axis, between the two charges, where the potential vanishes.
3. c) Find the point on the axis, to the right of $q_2$, where it also vanishes.
4. d) Reason why there is no point with $V = 0$ to the left of $q_1$.

Two charges $q_1 = 2\,\mu\text{C}$ and $q_2 = 3\,\mu\text{C}$, held by insulating supports, go from being separated by $1\ \text{m}$ to being separated by $0.25\ \text{m}$ (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Calculate the initial and final electric potential energy of the system.
2. b) Calculate the work that must be done by an external agent in the process, $W_{ext} = \Delta E_p$.
3. c) Reason the sign of the result: why must work be done to bring together two charges that repel each other?
4. d) What is the work done by the electric field in that same process? Relate it to that of the external agent.

In a hospital X-ray tube, electrons are accelerated towards the anode with a potential difference $\Delta V = 40\ \text{kV}$ (given: elementary charge $e = 1.6\times10^{-19}\ \text{C}$; electron mass $m_e = 9.11\times10^{-31}\ \text{kg}$; speed of light $c = 3\times10^{8}\ \text{m/s}$). The figure shows the diagram of the tube.

1. a) Express the potential difference in volts and calculate the kinetic energy of each electron on reaching the anode, in electronvolts and in joules.
2. b) Calculate the final speed of the electron using the classical expression $\frac{1}{2} \times m_e \times v^2 = e \times \Delta V$.
3. c) Compare the result with the speed of light and comment on why, at these energies, the classical approximation starts to become questionable.

Three identical charges $q = 1\ \text{nC}$ are placed at the vertices of an equilateral triangle of side length $10\ \text{cm}$ (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Calculate the potential energy of each pair of charges.
2. b) Add up the contributions from all pairs and calculate the total potential energy of the system.
3. c) Interpret the result as the total work required to "assemble" the distribution by bringing the charges, one by one, from infinity.
4. d) Reason why, in an arbitrary charge system, it is not sufficient to calculate the energy of one pair and multiply by the number of pairs: what condition is met in this case that makes it possible?

Your classmate claims: "if at a point the electric field is zero, the potential is also zero; and if the potential is zero, the field is also zero" (take $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$).

1. a) Refute the first part with a numerical example: find the point on the line segment between the charges $q_1 = 9\ \text{nC}$ and $q_2 = 4\ \text{nC}$, separated by $1\ \text{m}$, where the field is zero, and calculate the potential at that point.
2. b) Refute the second part with another numerical example: calculate the potential and the field at the midpoint between the charges $q_1 = +5\ \text{nC}$ and $q_2 = -5\ \text{nC}$, separated by $40\ \text{cm}$.
3. c) Explain, from the definitions of field (vector) and potential (scalar), why both statements are false in general.

In a region, the equipotential surfaces are parallel planes. Two of them, separated by $2\ \text{cm}$, have potentials $V_1 = 300\ \text{V}$ and $V_2 = 100\ \text{V}$. Observe the diagram:

1. a) Calculate the electric field between the two planes, assuming it to be uniform, $E = \frac{\Delta V}{d}$, and state its direction and sense (the field points towards decreasing potentials).
2. b) Calculate the work done by the field when moving a charge $q = 2\ \text{nC}$ from the plane of $300\ \text{V}$ to that of $100\ \text{V}$.
3. c) A student moves the charge first "sideways", without leaving the plane of $300\ \text{V}$, and only then carries it to the plane of $100\ \text{V}$ along a broken path. Reason, using the fact that the electric field is conservative (work depends only on the initial and final points), that the total work is the same as in part b), and calculate the work for the "sideways" section.

During a storm, the base of a cloud can be at a potential of about $10^{8}\ \text{V}$ relative to the ground, and a lightning bolt can transfer a charge of $20\ \text{C}$ (given: $1\ \text{kW}\times\text{h} = 3.6\times10^{6}\ \text{J}$).

1. a) Calculate the energy released in the lightning bolt, $E = Q \times \Delta V$.
2. b) Express it in kilowatt-hours.
3. c) The monthly electricity consumption of a household is about $200\ \text{kW}\times\text{h}$: compare both values and reason why, despite this huge amount of energy, it is so difficult to harness lightning energy (think about the instantaneous power and the unpredictability of the discharges).

Unit challenge. Design and calibrate a home-made oscillating electrostatic sensor to characterize the field created by an everyday object.

1. Setup and data collection: Build a pendulum with a thin thread and a small aluminum ball ($m \approx 0.05\text{ g}$). Measure the length $L \pm \Delta L$ with a ruler ($L = 0.300 \pm 0.001\text{ m}$). Time $10$ free oscillations with your phone (uncertainty $\pm 0.2\text{ s}$) to determine the natural period $T_0$.

2. Vector treatment and flux: Charge a plastic ruler by friction (model it as a uniform linear charge distribution by applying Gauss's law to approximate $\vec{E}$). Charge the ball by contact. When bringing the ruler close at a fixed distance, set up the vector equilibrium $\sum \vec{F} = \vec{T} + \vec{P} + q\vec{E} = \vec{0}$ from the deflection angle to express the net charge $q$ in terms of the ruler's linear charge density $\lambda$.

3. Oscillatory energy and report: Displace the pendulum with small amplitudes $A$ relative to the new equilibrium. Measure the resulting periods, obtain the equivalent restoring constant $k = m\omega^2$, and measure the maximum speed $v_{\max}$ in each case by video analysis; plot the mechanical energy $E_m = \frac{1}{2}m v_{\max}^2$ versus $A^2$ with its error bars and compare it with the theoretical line $\frac{1}{2}kA^2$. Summarize your conclusions in a short report.

Grading criteria:

• Rigor in the experimental treatment of data, propagation of uncertainties, and plotting.

• Correct application of Gauss's law and vector analysis of forces.

• Coherent justification of energy conservation in the oscillatory regime.