In the school laboratory, an old cathode ray oscilloscope shows a bright spot where a beam of electrons strikes its screen. Hold a bar magnet under the tube, with its north pole pointing up. The field then points up through the beam. The spot does not move up or down. It slides sideways. Which way does it go?
Three axes at right angles
A point on a line needs one number, and a point on a page needs two. A point in space needs three. Choose an origin O and three axes through it, each at right angles to the other two. The point P has the coordinates \((x, y, z)\), its distances along the three axes. Each pair of axes fixes a coordinate plane: the \(xy\) plane, where \(z = 0\), the \(yz\) plane and the \(zx\) plane. A floor is a plane \(z = 0\), and a shelf at \(2\,\text{m}\) is the plane \(z = 2\).
To find the distance of P from O, use Pythagoras twice. First walk \(x\) along the \(x\) axis and \(y\) along the \(y\) direction, in the floor, to the point Q under P. The floor gives \(OQ^2 = x^2 + y^2\). Then climb \(z\) straight up to P. The triangle OQP has a right angle at Q, so \(OP^2 = OQ^2 + z^2 = x^2 + y^2 + z^2\). The chapter Functions and Graphs uses the same formula between any two points.
A hall is \(12\,\text{m}\) long, \(4\,\text{m}\) wide and \(3\,\text{m}\) high. How long is the straight line from one corner on the floor to the opposite corner on the ceiling, in metres?
Right-handed axes
Three perpendicular axes can be labelled in two ways that are mirror images of each other. Physics always uses one of them, the right-handed axes. Curl the fingers of your right hand from the \(+x\) axis towards the \(+y\) axis. Your stretched thumb then points along \(+z\). In unit vectors this is \(\hat i \times \hat j = \hat k\), and the cycle goes on: \(\hat j \times \hat k = \hat i\) and \(\hat k \times \hat i = \hat j\).
Three perpendicular axes are right-handed when the fingers of the right hand, curled from \(+x\) towards \(+y\), leave the thumb pointing along \(+z\). For such axes \(\hat i \times \hat j = \hat k\), \(\hat j \times \hat k = \hat i\) and \(\hat k \times \hat i = \hat j\).
A page has only two directions, so the third must be drawn by a convention. With \(x\) to the right and \(y\) up the page, the right hand puts \(z\) out of the page, towards the reader. A vector out of the page is drawn \(\odot\), like the point of an arrow coming at you. A vector into the page is drawn \(\otimes\), like the tail feathers of an arrow flying away. An oblique drawing may instead show \(x\) to the right, \(y\) sloping into the page and \(z\) up. That set is right-handed too.
A figure draws the \(x\) axis to the right and the \(y\) axis up the page. Where must the \(z\) axis point for the axes to be right-handed?
Two rules from one hand
Physics uses the right hand in two ways. The first is for a product of two vectors, \(\vec A \times \vec B\). Point the fingers along \(\vec A\) and curl them towards \(\vec B\) through the smaller angle. The thumb then points along \(\vec A \times \vec B\). This gives the torque \(\vec\tau = \vec r \times \vec F\), the angular momentum \(\vec L = \vec r \times \vec p\) and the magnetic force \(\vec F = q\vec v \times \vec B\).
The second is the grip rule, for anything that turns or circulates. Curl the fingers of the right hand in the sense of the turning. The thumb then gives the direction of the axis vector. A wheel turning anticlockwise, seen from the front, has its angular velocity \(\vec\omega\) pointing out towards you. A current round a loop gives the loop's magnetic moment along the thumb. For a straight current, grip the wire with the thumb along the current, and the fingers curl the way the field lines go round it.
The two rules always agree. A particle that goes anticlockwise round O, in the page, has \(\vec L = \vec r \times \vec p\) out of the page by the product rule. Grip its turning with the fingers anticlockwise, and the thumb also points out of the page. Both rules are the right-handedness of the axes, used in two ways.
A ceiling fan turns anticlockwise when you look up at it from the floor. Which way does its angular velocity \(\vec\omega\) point?
Take \(x\) towards the east, \(y\) towards the north and \(z\) vertically up. A proton moves towards the north in a magnetic field that points vertically up. Which way is the magnetic force on it?
A flat coil lies on a table, and its current flows anticlockwise when seen from above. Which way does its magnetic moment point?
Back to physics: the force from components
When the vectors come in components, the cycle of the unit vectors does all the work. Each product of two different unit vectors is the third, with a plus sign along the cycle and a minus sign against it. The chapter Magnetic Force on Moving Charges works one such force.
A charge of \(2\,\mu\text{C}\) moves with \(\vec v = (3\hat i + 4\hat j) \times 10^4\,\text{m/s}\) in a field \(\vec B = 0.1\hat k\,\text{T}\). Find the magnetic force and its size.
\(\vec F = q\,\vec v \times \vec B\) with \(q = 2 \times 10^{-6}\,\text{C}\). The field is along \(\hat k\), so only \(\hat i \times \hat k\) and \(\hat j \times \hat k\) are needed.
\(\vec v \times \vec B = 10^4 \times 0.1\,(3\,\hat i \times \hat k + 4\,\hat j \times \hat k) = 10^3\,(-3\hat j + 4\hat i)\).
\(\vec F = 2 \times 10^{-6} \times 10^3\,(4\hat i - 3\hat j) = (8\hat i - 6\hat j) \times 10^{-3}\,\text{N}\). Its size is \(\sqrt{64 + 36} \times 10^{-3} = 0.01\,\text{N}\).
The force is \((8\hat i - 6\hat j) \times 10^{-3}\,\text{N}\), of size \(0.01\,\text{N}\), the answer of the physics chapter. It lies in the \(xy\) plane, at right angles to the field.
The oscilloscope has its answer too. Take \(x\) along the beam, towards the screen, and \(z\) up, along the field. For right-handed axes \(y\) then points to the left as you face the screen. For a positive charge \(\vec v \times \vec B\) would lie along \(\hat i \times \hat k = -\hat j\), to the right. An electron is negative, so its force points the other way. The spot slides to the left.
Symmetry: answers without sums
Some answers in space come from looking, not from calculating. Suppose a body looks exactly the same after a turn, a reflection or a shift. Then its field, its flux or its current must look the same after that change too. This is a symmetry argument. It never gives a number by itself, but it tells which directions are possible and which parts must be equal.
Let a charge, a current or a network look the same after a rotation, a reflection or a shift. Then whatever it produces, such as a field, a flux or a current, is unchanged by the same operation.
A uniformly charged sphere looks the same after any turn about its centre. So its field must point straight out, along the radius, with one size at one distance. A long straight charged line looks the same after a turn about it and a shift along it. So its field points straight out from the line. That is why Gauss's law takes a sphere round a sphere and a cylinder round a line. A charge at the centre of a cube sees six faces that look alike, so each face carries a sixth of the flux.
Twelve equal resistors form the edges of a cube. A current of \(6\,\text{A}\) enters at one corner and leaves at the opposite corner. How much current flows along each of the three edges that meet at the entry corner, in amperes?
The right-hand rule gives \(\vec v \times \vec B\), which is the force on a positive charge. For an electron, reverse it. A figure drawn with \(z\) into the page, while \(x\) points right and \(y\) up, is left-handed. Every cross product in it comes out backwards. Fleming's left-hand rule is a separate aid for motors. Use one rule throughout a problem, never both at once.
Motion in Two Dimensions writes the equations of motion as vectors with a third axis. Kinematics of Circular Motion points the angular velocity along the axis, and Torque and Equilibrium and Angular Momentum find the directions of \(\vec r \times \vec F\) and \(\vec r \times \vec p\). Electric Flux and Gauss's Law chooses its surfaces by symmetry and shares the flux of a charge among the faces of a cube. Circuits: Kirchhoff's Laws, Networks and RC Transients finds the resistance of a cube of resistors by symmetry. The three chapters on magnetism, from Magnetic Force on Moving Charges to Sources of the Magnetic Field, find every magnetic direction with the right hand.
- Drawing \(z\) into the page when \(x\) points right and \(y\) points up.Right-handed axes put \(z\) out of the page. Check with the fingers from \(+x\) to \(+y\).
- Curling the fingers from the second vector to the first.For \(\vec A \times \vec B\), point along \(\vec A\) first and curl towards \(\vec B\). Reversing the order reverses the answer.
- Forgetting the negative charge of an electron.The rule gives \(\vec v \times \vec B\). Multiply by the sign of the charge, so an electron's force points the other way.