Gravitational, Electric, and Magnetic Fields: Test Review

Everything for the Unit 3 test on one page: the key facts from all 6 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Universal Gravitation and Orbits

Universal gravitation: $F_{g} = \frac{G m _{1} m _{2}}{r ^{2}}$ , with $r$ measured centre to centre and $G = 6.67 \times 1 0^{- 11} N \cdot m^{2} / kg^{2}$ .
Gravitational field strength: $g = \frac{F _{g}}{m} = \frac{GM}{r ^{2}}$ , in $N/kg$ .
For a circular orbit gravity is the centripetal force, so $v = \frac{GM}{r}$ and $T = \frac{2 π r}{v}$ .
Orbital radius is Earth's radius plus altitude: $r = R_{E} + h$ . The satellite's own mass does not affect its speed or period.
A geostationary satellite orbits above the equator with a period of one day, at $r \approx 4.22 \times 1 0^{7} m$ .
Apparent weightlessness happens when an object and its surroundings share the same free-fall acceleration, so no support force acts.

Gravitational Potential Energy, Escape Speed and Kepler's Laws

Gravitational potential energy with zero at infinity: $E_{p} = - \frac{GM m}{r}$ ; it is negative because the object is bound.
Change in potential energy between two radii: $Δ E_{p} = GM m (\frac{1}{r _{1}} - \frac{1}{r _{2}})$ , which reduces to $m g h$ near the surface.
In a circular orbit, $E_{k} = \frac{GM m}{2 r}$ and $E_{t o t a l} = - \frac{GM m}{2 r}$ ; the binding energy is $\frac{GM m}{2 r}$ .
Escape speed: $v_{esc} = \frac{2 GM}{r} = 2 v_{or bi t}$ , independent of the object's mass and launch direction.
Kepler's laws: elliptical orbits with the Sun at one focus; equal areas in equal times; $T^{2} \propto r^{3}$ , so $\frac{T _{1}^{2}}{r _{1}^{3}} = \frac{T _{2}^{2}}{r _{2}^{3}}$ for the same central body.

Electric Forces and Fields

Coulomb's law: $F_{e} = \frac{k q _{1} q _{2}}{r ^{2}}$ with $k = 8.99 \times 1 0^{9} N \cdot m^{2} / C^{2}$ . Like charges repel; unlike charges attract.
Superposition: the net force (or field) from several charges is the vector sum of the individual forces (or fields).
Electric field strength: $E = F_{e} / q$ in N/C, directed as the force on a positive test charge.
Field of a point charge: $E = \frac{k q}{r ^{2}}$ , pointing away from positive and toward negative charges.
Between large parallel plates the field is uniform; around a point charge it is radial and inverse square.

Electric Potential Energy and Potential

Electric potential energy of two charges: $E_{e} = \frac{k q _{1} q _{2}}{r}$ , zero at infinite separation, used with signs.
Electric potential: $V = E_{e} / q$ and, for a point charge, $V = \frac{k q}{r}$ . Potentials add as scalars.
Work to move a charge between two points: $W = q Δ V$ .
Uniform field between parallel plates: $E = \frac{Δ V}{d}$ , in V/m, which equals N/C.
A charge accelerated from rest through $Δ V$ gains $E_{k} = ∣ q ∣Δ V$ ; $1 eV = 1.60 \times 1 0^{- 19} J$ .
Millikan's balanced drop: $qE = m g$ , so $q = \frac{m g d}{Δ V}$ ; every charge is a whole-number multiple of $e$ .

Magnetic Fields and Forces

Force on a moving charge: $F = q v B sin θ$ , maximum when $v$ is perpendicular to $B$ and zero when parallel.
Right-hand rule 3 (positive charge or conventional current): fingers along $B$ , thumb along $v$ or $I$ , palm gives the force. Reverse the direction for a negative charge.
In a uniform field a charge moving perpendicular to $B$ circles with radius $r = \frac{m v}{qB}$ . The magnetic force does no work.
Force on a current-carrying wire: $F = B I L sin θ$ .
Right-hand rules 1 and 2: thumb along the current in a straight wire gives the circular field; fingers along the current in a coil give the north end at the thumb.
Velocity selector: only charges with $v = E / B$ pass straight through crossed electric and magnetic fields.

Charged Particles in Uniform Fields: Deflection and e/m

Electron gun: $q Δ V = \frac{1}{2} m v^{2}$ , so $v = 2 q Δ V / m$ for a charge starting from rest.
Between parallel plates a charge has constant velocity along the plates and constant acceleration $a = qE / m = q Δ V / (m d)$ across them, so its path is a parabola.
Treat deflection like a horizontal projectile: $t = L / v_{x}$ , $y = \frac{1}{2} a t^{2}$ , and $tan θ = v_{y} / v_{x}$ on exit.
Velocity selector: crossed fields pass undeflected only the particles with $v = E / B$ .
Thomson's method: $\frac{e}{m} = \frac{v}{B r} = \frac{E}{B ^{2} r}$ ; the accepted value is $1.76 \times 1 0^{11} C/kg$ .
Electric and gravitational fields do work and bend paths into parabolas; a magnetic field does no work and bends paths into circles.

Where marks get lost

Using the altitude as $r$ . The distance in $F_{g} = G m_{1} m_{2} / r^{2}$ and $v = GM / r$ is measured from Earth's centre, so add Earth's radius first. (Universal Gravitation and Orbits)
Forgetting to square $r$ , or halving the force when the distance doubles. It is an inverse square: doubling $r$ quarters the force. (Universal Gravitation and Orbits)
Putting the satellite's mass into $v = GM / r$ . $M$ is the mass being orbited; the satellite's mass cancels. (Universal Gravitation and Orbits)
Saying there is no gravity in orbit. Gravity in low Earth orbit is nearly 90% of its surface value; it is the support force that is missing. (Universal Gravitation and Orbits)
Leaving the period in hours or days when using $G$ in SI units. Convert to seconds, then convert back at the end. (Universal Gravitation and Orbits)
Dropping the minus sign in $E_{p} = - GM m / r$ , which makes moving outward look like a loss of potential energy. Moving away from a planet always increases (makes less negative) the potential energy. (Gravitational Potential Energy, Escape Speed and Kepler's Laws)
Using the altitude instead of the distance from the centre. Every $r$ in $- GM m / r$ , $2 GM / r$ and Kepler's third law is measured from the centre of the central body. (Gravitational Potential Energy, Escape Speed and Kepler's Laws)
Thinking escape speed depends on the mass of the object or the direction it is launched. Mass cancels and energy is a scalar, so neither matters (as long as the path misses the planet). (Gravitational Potential Energy, Escape Speed and Kepler's Laws)
Assuming a satellite in a higher orbit moves faster because it has more energy. Its total energy is higher, but its kinetic energy and speed are lower. (Gravitational Potential Energy, Escape Speed and Kepler's Laws)
Applying $T^{2} / r^{3}$ ratios between objects orbiting different central bodies, such as the Moon and Mars. The constant $4 π^{2} / (GM)$ depends on the mass being orbited. (Gravitational Potential Energy, Escape Speed and Kepler's Laws)
Forgetting to convert microcoulombs ( $1 0^{- 6}$ ), nanocoulombs ( $1 0^{- 9}$ ) or centimetres before substituting. One missed conversion changes the answer by factors of a thousand or more. (Electric Forces and Fields)
Using the signs of the charges to set the direction of the force in a multi-charge problem. Find magnitudes, then decide each direction from attract or repel. (Electric Forces and Fields)
Adding the magnitudes of forces that point in different directions. Forces and fields are vectors; add components. (Electric Forces and Fields)
Thinking the field direction is the direction any charge moves. $E$ is defined by the force on a positive test charge; electrons are pushed the opposite way. (Electric Forces and Fields)
Including the test charge's own field when finding the field at its location. The field at a point comes from the other charges only. (Electric Forces and Fields)
Squaring $r$ in $E_{e} = k q_{1} q_{2} / r$ or $V = k q / r$ . Only the force and field use $r^{2}$ . (Electric Potential Energy and Potential)
Adding potentials from several charges as vectors, or dropping their signs. Potential is a scalar: add with signs, no components. (Electric Potential Energy and Potential)
Leaving the plate separation in centimetres or millimetres in $E = Δ V / d$ . Convert to metres, or the field is off by a factor of 100 or 1000. (Electric Potential Energy and Potential)
Putting an energy in electron-volts straight into $\frac{1}{2} m v^{2}$ with a mass in kilograms. Convert eV to joules first. (Electric Potential Energy and Potential)
Confusing $E$ (field strength in N/C or V/m) with $E_{e}$ or $E_{k}$ (energies in joules). (Electric Potential Energy and Potential)
Using $cos θ$ instead of $sin θ$ , or measuring $θ$ from the wrong line. The angle is between the velocity (or wire) and the field, and a charge moving along the field feels no force. (Magnetic Fields and Forces)
Applying right-hand rule 3 to an electron without reversing the answer. The rule gives the force on a positive charge. (Magnetic Fields and Forces)
Thinking the magnetic force speeds a charge up. It is always perpendicular to the velocity, so it does no work and only changes the direction. (Magnetic Fields and Forces)
Expecting a stationary charge to feel a magnetic force. Only moving charges do. (Magnetic Fields and Forces)
Mixing up the fingers and thumb among the three right-hand rules. For a straight wire the thumb is the current; for a coil the fingers are the current. (Magnetic Fields and Forces)
Using the plate separation $d$ as the distance travelled along the plates. The time between the plates comes from the plate length $L$ and the horizontal speed; $d$ only sets the field $E = Δ V / d$ . (Charged Particles in Uniform Fields: Deflection and e/m)
Letting the horizontal speed change between the plates. The electric force is perpendicular to the plates, so the component along them stays constant, exactly as $v_{x}$ does for a projectile. (Charged Particles in Uniform Fields: Deflection and e/m)
Continuing the parabola after the particle leaves the plates. Outside the field there is no force, so the path is a straight line at the exit angle. (Charged Particles in Uniform Fields: Deflection and e/m)
Writing $e / m = E / (B r)$ instead of $E / (B^{2} r)$ : the field $B$ appears once in $v = E / B$ and again in $r = m v / (e B)$ . (Charged Particles in Uniform Fields: Deflection and e/m)
Expecting a magnetic field to speed a particle up or slow it down. The magnetic force is always perpendicular to the velocity, so it changes only the direction. (Charged Particles in Uniform Fields: Deflection and e/m)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Universal Gravitation and Orbits
1
The distance between the centres of two asteroids triples. What happens to the gravitational force between them?
Universal Gravitation and Orbits
2
What is the orbital speed of a satellite $800 km$ above Earth's surface, in km/s? Give three significant figures.
km/s
Universal Gravitation and Orbits
3
Which statement best explains why astronauts float inside a spacecraft in low Earth orbit?
Universal Gravitation and Orbits
4
Order these Earth orbits from shortest period to longest period.
1. 1The Moon's orbit, about $384000 km$ from Earth
2. 2A geostationary orbit, about $35900 km$ above the surface
3. 3A circular orbit $20000 km$ above the surface
4. 4A circular orbit $400 km$ above the surface
5. 5A circular orbit $2000 km$ above the surface
Gravitational Potential Energy, Escape Speed and Kepler's Laws
5
What is the escape speed from the surface of the Moon ( $M = 7.35 \times 1 0^{22} kg$ , $R = 1.74 \times 1 0^{6} m$ ), in km/s? Give three significant figures.
km/s
Gravitational Potential Energy, Escape Speed and Kepler's Laws
6
How much energy must be added to move a $1000 kg$ satellite from a circular orbit of radius $7.0 \times 1 0^{6} m$ to one of radius $1.4 \times 1 0^{7} m$ , in gigajoules? Use $M_{E} = 5.97 \times 1 0^{24} kg$ and give three significant figures.
GJ
Gravitational Potential Energy, Escape Speed and Kepler's Laws
7
Two probes, one of $100 kg$ and one of $2000 kg$ , are launched from Earth's surface. How do their escape speeds compare?
Gravitational Potential Energy, Escape Speed and Kepler's Laws
8
Match each of Kepler's laws or results to what it states.
First law
Second law
Third law
Constant in the third law
Electric Forces and Fields
9
Two small spheres carry charges of $+ 1.5 μ C$ and $+ 2.5 μ C$ and are $5.0 cm$ apart. What is the size of the electric force between them, in newtons? Give three significant figures.
N
Electric Forces and Fields
10
Which way do the electric field lines point near an isolated negative point charge?
Electric Forces and Fields
11
What is the name of the principle that says the net force on a charge is the vector sum of the forces from each other charge acting alone?
Electric Forces and Fields
12
Order the steps for finding the net electric force on a charge due to two other charges that are not in a line.
1. 1Find the resultant's magnitude and direction from the component sums
2. 2Resolve the forces into x- and y-components and add them
3. 3Decide each force's direction from attraction or repulsion
4. 4Convert all charges and distances to coulombs and metres
5. 5Find the size of each force with Coulomb's law using magnitudes
Electric Potential Energy and Potential
13
Two parallel plates $4.0 mm$ apart are connected to a $120 V$ supply. What is the electric field strength between them, in V/m?
V/m
Electric Potential Energy and Potential
14
Which of these is equal to one electron-volt?
Electric Potential Energy and Potential
15
What name is given to a line joining points that are all at the same electric potential?
Electric Potential Energy and Potential
16
What was the most important conclusion of Millikan's oil-drop experiment?
Magnetic Fields and Forces
17
A $0.25 m$ wire carrying $8.0 A$ lies perpendicular to a $0.15 T$ magnetic field. What is the magnetic force on the wire, in newtons?
N
Magnetic Fields and Forces
18
A charged particle moves in a circle in a uniform magnetic field. How much work does the magnetic force do on it during one revolution?
Magnetic Fields and Forces
19
An electron moves toward the right side of the page through a uniform magnetic field directed into the page. Which way does the magnetic force on it point?
Magnetic Fields and Forces
20
These particles all enter the same magnetic field at the same speed, perpendicular to the field. Order their circular paths from smallest radius to largest.
1. 1A proton
2. 2A singly charged ion with 20 times the mass of a proton
3. 3An alpha particle (two protons and two neutrons, charge $+ 2 e$ )
4. 4An electron
Charged Particles in Uniform Fields: Deflection and e/m
21
An electron starts from rest and accelerates through a potential difference of $500 V$ . What is its final speed, in units of $1 0^{7} m/s$ ? Give three significant figures.
× 10⁷ m/s
Charged Particles in Uniform Fields: Deflection and e/m
22
A velocity selector uses $E = 4.5 \times 1 0^{4} N/C$ and $B = 3.0 \times 1 0^{- 3} T$ . What speed do the undeflected particles have, in km/s?
km/s
Charged Particles in Uniform Fields: Deflection and e/m
23
In a velocity selector, a particle moves faster than $E / B$ . What happens to it?
Charged Particles in Uniform Fields: Deflection and e/m
24
Match each technology to how it uses fields to control charged particles.
Oscilloscope
Continuous inkjet printer
Ion thruster
Mass spectrometer

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…