# Scalar and Vector Quantities

#### Red_CCF

Hi

Just wondering if someone can tell me if the following are scalar or vector quantities and why

Current
Potential
Potential Difference

Also, I'm wondering if we include plus/minus signs in calculations depending on the charge. Ex. would current be negative if it was a negative charge moving.

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#### InTuoVultu

Current: depends on the context. In the general sense, vector. Because we have charge that can flow at any rate in any direction. However, if you're dealing with circuits, where you know the direction (it's only flowing along the wire) then it's fine just to specify the magnitude. Usually in amps (coulombs/second)

Electric Potential: scalar. The classic analogy here is balls rolling on hills. Whatever book you're using probably uses this analogy. If it doesn't it's a bad book. If you have some shape of hill, you can assign potential energy values to every point on that hill. You're energy is going to be something proportional to the height. (remember pot.E.=m g h). This energy is a scalar because it's energy. Now it's the same for E fields as gravity. Every point in space has an energy associated with it just like the balls did. Only with a few twists. Firstly, the energy that an object has at any one place is proportional to whatever the charge on that object is. This is just like the mass in mgh. Only it's not always useful to carry this around so we divide it out.

pot.E.
_____ = gh
m

This pot.E/m is the equivalent of the electric potential. Multiply any electric potential by a charge and you get the potential energy of an object with that charge at that point. Potential and voltage are used pretty interchangeably.

Potential difference is also a scalar. Here we're talking about the difference in potentials between two points. As in, "at (1,1,0) the potential is 5 J/C and at (1,2,0) the potential is 3 J/C so the potential difference between the two points is 2 J/C"

hope that helps. Remember energy is always a scalar.

#### Red_CCF

Current: depends on the context. In the general sense, vector. Because we have charge that can flow at any rate in any direction. However, if you're dealing with circuits, where you know the direction (it's only flowing along the wire) then it's fine just to specify the magnitude. Usually in amps (coulombs/second)

Electric Potential: scalar. The classic analogy here is balls rolling on hills. Whatever book you're using probably uses this analogy. If it doesn't it's a bad book. If you have some shape of hill, you can assign potential energy values to every point on that hill. You're energy is going to be something proportional to the height. (remember pot.E.=m g h). This energy is a scalar because it's energy. Now it's the same for E fields as gravity. Every point in space has an energy associated with it just like the balls did. Only with a few twists. Firstly, the energy that an object has at any one place is proportional to whatever the charge on that object is. This is just like the mass in mgh. Only it's not always useful to carry this around so we divide it out.

pot.E.
_____ = gh
m

This pot.E/m is the equivalent of the electric potential. Multiply any electric potential by a charge and you get the potential energy of an object with that charge at that point. Potential and voltage are used pretty interchangeably.

Potential difference is also a scalar. Here we're talking about the difference in potentials between two points. As in, "at (1,1,0) the potential is 5 J/C and at (1,2,0) the potential is 3 J/C so the potential difference between the two points is 2 J/C"

hope that helps. Remember energy is always a scalar.
What about in AC circuits where potential difference goes in opposite directions? I've seen this as an argument that voltage should be vector. Also, posters on PF suggests that I use the signs of the charges (negative sign for negative charges) in calculating potential, wouldn't that make it a vector too? Thanks.

#### Fightfish

Current: depends on the context. In the general sense, vector. Because we have charge that can flow at any rate in any direction. However, if you're dealing with circuits, where you know the direction (it's only flowing along the wire) then it's fine just to specify the magnitude. Usually in amps (coulombs/second)
I don't quite agree with that though. Current in its true form is just the rate of flow of electric charge, and hence it is a scalar. In formalism of the force on a current-carrying conductor or even the Biot-Savart law, it is evident that current is a scalar since a $$d\vec{l}$$ construct is used to represent the direction of current flow.

#### Hootenanny

Staff Emeritus
Gold Member
I don't quite agree with that though. Current in its true form is just the rate of flow of electric charge, and hence it is a scalar. In formalism of the force on a current-carrying conductor or even the Biot-Savart law, it is evident that current is a scalar since a $$dl$$ construct is used to represent the direction of current flow.
I completely agree, current most certainly is a scalar. In addition to the examples already provided, one can most easily see that current must be a scalar through it's relation to current density, namely

$$I = \boldsymbol{J\cdot A}$$

#### Red_CCF

I completely agree, current most certainly is a scalar. In addition to the examples already provided, one can most easily see that current must be a scalar through it's relation to current density, namely

$$I = \boldsymbol{J\cdot A}$$
That's what I was thinking too. But someone who argues in favour of current being a vector says that in an AC circuit current goes both ways so direction matters when we put a diode on the circuit. Can someone provide a counter-arguement to this?

#### dx

Homework Helper
Gold Member
Charge can go both ways through a surface, but you don't need a vector to represent the current I. Scalars can be positive or negative.

#### jambaugh

Gold Member
Ultimately when one is distinguishing "scalar" vs. "vector" vs. "spinor" etc one is speaking with respect to a particular group. So one can speak of the current through a surfaces as a "scalar" with respect to SO(3) spatial rotations and as a "vector" with respect to an abstract O(1) current reversing group.

I like to start my students thinking of signed quantities as "kinda like vectors" including, say, signed areas representing integrals to emphasize their additive properties. There is no harm in this and it can help with keeping signs straight.

#### dx

Homework Helper
Gold Member
... as a "vector" with respect to an abstract O(1) current reversing group.
The current through a surface is defined as ∫S <j,dA>. How can that be a vector?

#### Red_CCF

Charge can go both ways through a surface, but you don't need a vector to represent the current I. Scalars can be positive or negative.
But I thought that scalar only meant the magnitude, so positive only???

#### dx

Homework Helper
Gold Member
But I thought that scalar only meant the magnitude, so positive only???
No, the standard meaning of the word 'scalar' is 'real number'. The magnitude is usually called 'absolute value'.

#### Red_CCF

No, the standard meaning of the word 'scalar' is 'real number'. The magnitude is usually called 'absolute value'.
Can you give me an example of a scalar quantity that can be negative? The ones I can think of (mass, time, energy etc) all must be positive

#### Andy Resnick

I completely agree, current most certainly is a scalar. In addition to the examples already provided, one can most easily see that current must be a scalar through it's relation to current density, namely

$$I = \boldsymbol{J\cdot A}$$
Charge is a 3-form (See, for example, MTW, Gravitation, p113-4), so current density is a vector.

#### Hootenanny

Staff Emeritus
Gold Member
Can you give me an example of a scalar quantity that can be negative? The ones I can think of (mass, time, energy etc) all must be positive
Energy and time can both be negative.
Charge is a 3-form (See, for example, MTW, Gravitation, p113-4), so current density is a vector.
I never said current density wasn't a vector.

#### Red_CCF

Energy and time can both be negative.
Really? Energy I could understand but time?

EDIT: I read that if force is applied in the opposite direction of movement then energy is negative. But is there any way to calculate it? For instance, how do we determine the distance that the force going int he opposite direction applied for.

#### jambaugh

Gold Member
The current through a surface is defined as ∫S <j,dA>. How can that be a vector?
A vector is an element of a vector space. Its a mathematical concept. Scalars are also "vectors" in a one dimensional "vector space".

Again read my post. What makes something a "vector", "scalar", "tensor" et al is not an absolute but rather defined by how it transforms relative to a particular group. Any set of (signed) quantities can be called a "vector" by choosing the appropriate group of transformations (though it may be a silly, useless example) and any quantity may be defined as a "scalar" but again so defined relative to a choice of transformation group (which leaves them invariant).

But referring to a quantity as "vector" or "scalar" without giving or at least implying the contextual group of transformations isn't proper. It's like saying a particle has mass 4 without giving the units. What? 4 kilograms? 4 slugs? 4 solar masses?

So yes in an appropriate context current can be considered a (one dimensional) "vector". In another context the components of a 3-force can be considered "scalars" (e.g. with respect to the quark color gauge group SU(3) ). It's all relative.

#### jambaugh

Gold Member
Really? Energy I could understand but time?

EDIT: I read that if force is applied in the opposite direction of movement then energy is negative. But is there any way to calculate it? For instance, how do we determine the distance that the force going int he opposite direction applied for.
No that's not right. If force opposes the direction of motion then the work done on the object is negative. That's a net change in energy. BTY Time can certainly be negative, (3 weeks ago = -3weeks). Or e.g. 3000BC = -3000years. Time is a coordinate and is meaningful only relative to a choice of origin (zero value). Note that the countdown clock for a launch is expressing negative values of time... hence the magnitude decreases as time moves forward.

#### Red_CCF

If force opposes the direction of motion then the work done on the object is negative. That's a net change in energy.
Yeah that's what I meant. If an object is moving and work is done against the object, how do we calculate work because how we find the distance aspect is quite confusing to me.

Also, when we are calculating potential, current, etc. do we include the sign of the charge? Ex. if we are finding out the potential of a negative charge, do we include the negative sign of the charge in our calculation?

#### jambaugh

Gold Member

Yeah that's what I meant. If an object is moving and work is done against the object, how do we calculate work because how we find the distance aspect is quite confusing to me.
Work done on object = (vector)Force (dot) (vector)Displacement.
Note if two vectors are near opposite directions (angle between them is > 90deg) then their dot product is negative.

Also, when we are calculating potential, current, etc. do we include the sign of the charge? Ex. if we are finding out the potential of a negative charge, do we include the negative sign of the charge in our calculation?
Yes. A stream of electrons flowing into a volume is a net current out of that volume since the electrons have negative charge.

Also note that charge times potential gives potential energy. Given say an electron and a proton, the potential energy is negative (binding them together) which you can calculate either as the negative charge of the electron times the positive potential from the proton or by the positive charge of the proton times the negative potential from the electron.

I say that the negative potential binds them together because as you pull them apart the potential decreases in magnitude. With opposite charges you get potential which is negative in sign and this means it is actually increasing in value (toward zero from below) as they are drawn apart. Thus we see the manifestation of opposite charges attracting (and like charges repelling) in the sign convention.

#### Red_CCF

Work done on object = (vector)Force (dot) (vector)Displacement.
Note if two vectors are near opposite directions (angle between them is > 90deg) then their dot product is negative.
I actually haven't gotten to the point where I apply linear algebra into physics. I'm just basically thinking about it using simple kinematics still. Like if we have a car travelling at constant velocity and we apply a force to stop it, how do we know how much work is done knowing the amount of force that is applied? But if the answer involves dot products of vectors than I think I'll just leave it for now.

I say that the negative potential binds them together because as you pull them apart the potential decreases in magnitude. With opposite charges you get potential which is negative in sign and this means it is actually increasing in value (toward zero from below) as they are drawn apart. Thus we see the manifestation of opposite charges attracting (and like charges repelling) in the sign convention.
But what about in a circuit, where an electrons goes from a low potential to high potential but its potential energy goes from high to low? In terms of magnitude only an electron's potential would be decreasing but if we include the signs of charges then it's entirely different.

Thanks for any help that you can provide

#### jambaugh

Gold Member
I actually haven't gotten to the point where I apply linear algebra into physics. I'm just basically thinking about it using simple kinematics still. Like if we have a car travelling at constant velocity and we apply a force to stop it, how do we know how much work is done knowing the amount of force that is applied? But if the answer involves dot products of vectors than I think I'll just leave it for now.
Don't be too intimidated.

The dot product of two vectors A and B is the product of the magnitudes times the cosine of the angle.

Thus if they are at 90deg the cosine is zero and the dot product is zero.

If they are in the same direction (angle = 0) the cosine is 1 and the dot product is just the product of their magnitudes.

If they are in opposite direction (angle = 180deg) then the cosine is -1 and the dot product is just the negative of the product of the magnitudes.

This last is the case for the braking car you mentioned.

It's then a short step to using trig. to get general dot products for vectors at arbitrary angles. But you can also calculate it by adding up the products of components.

With regard to potentials it will help to explicitly distinguish electrical potential (volts) and potential energy (joules). The current always flows so as to reduce potential energy (water flows down-hill). Electrons do this by flowing toward higher electrical potential (anode) while positive charged particles do this by flowing toward lower electrical potential (cathode). I've added qualifiers to your quoted question below in red:

But what about in a circuit, where an electrons goes from a low [electrical] potential to high [electrical] potential but its potential energy goes from high to low?
Right it flows "downhill" with respect to potential energy. To do so it must flow "uphill" with respect to electrical potential. Think of the negative charge of electrons as making them "boyant" so they "rise" toward the positive potential. Note a rising bubble (in say beer) effects a net downard mass flow since (heavier) beer is filling in the space lower down that the bubble occupied earlier. (So next time you watch a glass of Guiness forming its head imagine the bubbles as little electrons.)
In terms of magnitude only an electron's potential would be decreasing but if we include the signs of charges then it's entirely different.
Thanks for any help that you can provide
I'm not sure which potential you mean here and magnitude is relative to the choice of zero (ground) in a circuit so it isn't quite unambiguous as you stated it.

An electron may flow from a point at -120 volts to 0 volts or from a point at 20 volts to one at 140volts. The difference is only where one (arbitrarily) chooses to affix the ground i.e. pick a point where the voltage is defined to be zero. But this choice messes up what you said about magnitudes. In each case the change in electrical potential is +120 volts and the change in energy is -e * 120 joules where -e is the electron charge in Coulombs (Coulombs times Volts = Joules). But note in one case magnitude decreases and in the other magnitude increases.

(In the example I gave in an earlier post I was referring to free particles with an inverse square potential. The convention is to let the potential be zero at infinity but this is again a little bit arbitrary.)

Note that in a simple battery circuit the current flows from the positive terminal to the negative terminal but the actual electrons move in the opposite direction.

This is due to an unfortunate accident of history. We can blame (I'm pretty sure) Benjamin Franklin. At the time it wasn't known what actually carried charge except it was called electricity. I think Franklin picked the sign based on how he was producing the electrostatic charges. Probabily he was "charging up" a rod by rubbing it with a cloth which actually rubbed off spare electrons from the surface. When he found different materials produced an charge which canceled out the first case he had to pick one to be +charge and one to be -charge.

It wasn't until Edison and Thompson observed charges flowing off of hot filaments that we realized the principle charge carriers (electrons) had been taged with a negative sign.

We forgive Franklin this little "mistake" and acknowledge his brilliant insight into the nature of electricity. But if only we could go back and change all the text books it would be nice to reverse the convention so the electrons charge would be "positive".

#### dx

Homework Helper
Gold Member
A vector is an element of a vector space. Its a mathematical concept. Scalars are also "vectors" in a one dimensional "vector space".
I am aware of the idea of an abstract vector space, but real number scalars in physics (charge, mass, temperature etc.) are never called vectors.

Again read my post. What makes something a "vector", "scalar", "tensor" et al is not an absolute but rather defined by how it transforms relative to a particular group. Any set of (signed) quantities can be called a "vector" by choosing the appropriate group of transformations (though it may be a silly, useless example) and any quantity may be defined as a "scalar" but again so defined relative to a choice of transformation group (which leaves them invariant).
First of all, elements of vector spaces "transform" only once you have given a representation of them in a coordinate sytem. This representation need not be covariant with respect to Lorentz transformations or whatever, but it has no bearing on whether the original set is a vector space. The hyperbola representation (xaxa = -1) of the Minkowski metric in an inertial frame is covariant with respect to Lorentz transformations (i.e. it has Lorentz symmetry), but it is not covariant with respect to general coordinate transformations. This has no bearing on whether the original object which is being represented, a tensor, belongs to a vector space. The transformation group which acts as a symmetry of the representation depends on the representation itself. Euclidean geometry can be represented in an orthogonal coordinate system by the dot product a1b1 + a2b2 + a3b3; this representation has a symmetry group O(3). This same object (the euclidean metric) can be represented in a different way (analogous to the metric in special relativity) which has a symmetry GL(3). So we have changed the symmetry group by changing the representation, but this has no effect on the tensor character of the metric.

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#### Red_CCF

Right it flows "downhill" with respect to potential energy. To do so it must flow "uphill" with respect to electrical potential. Think of the negative charge of electrons as making them "boyant" so they "rise" toward the positive potential. Note a rising bubble (in say beer) effects a net downard mass flow since (heavier) beer is filling in the space lower down that the bubble occupied earlier. (So next time you watch a glass of Guiness forming its head imagine the bubbles as little electrons.)
I didn't really get that analogy and am still quite confused on why potential increases while potential energy decreases if they are proportional to each other. So the rising bubble is the electrons and the surface is the positive terminal. What is the heavy beer filling the space down analogous to?

Also, I'm wondering whether to include the positive/negative signs of charges in current, voltage, and potential calculations because I've seen some that do and some that don't.

Thanks.

#### Hootenanny

Staff Emeritus
Gold Member
I didn't really get that analogy and am still quite confused on why potential increases while potential energy decreases if they are proportional to each other. So the rising bubble is the electrons and the surface is the positive terminal. What is the heavy beer filling the space down analogous to?
The concepts of potential and potential energy are often quite difficult to grasp as first. The electric potential $\varphi$ and the potential energy are linked by the simple linear relation,

$$U = q\varphi$$

So the change in potential and potential energy for an electron (q=-e) is,

$$\Delta U = -e\Delta\varphi$$

Therefore, if $\Delta\varphi > 0$, then $\Delta U < 0$. The negative sign is most crucial. Obviously, for a positive charge the opposite would be true.
Also, I'm wondering whether to include the positive/negative signs of charges in current, voltage, and potential calculations because I've seen some that do and some that don't.
There are some cases where you may safely ignore negative signs, e.g. because they cancel. However, these are short-cuts and are not generally applicable. If you are not sure, it is best to include all signs in your calculations.

#### jambaugh

Gold Member
I didn't really get that analogy and am still quite confused on why potential increases while potential energy decreases if they are proportional to each other. So the rising bubble is the electrons and the surface is the positive terminal. What is the heavy beer filling the space down analogous to?

Also, I'm wondering whether to include the positive/negative signs of charges in current, voltage, and potential calculations because I've seen some that do and some that don't.

Thanks.
Remember two quantities are proportional if there's a constant k so one is k times the other. The proportionality constant can be a negative number. Think of property taxes. You have to pay out (negative income) in proportion to how much you own (positive asset) as compared to interest on an investment (positive income) in proportion to how much you own (positive asset).

The heavy beer is not analogous to anything in the electrical case. It is necessary because we don't have negative masses per se. I used it to create negative effective masses for the bubbles.

Let me try the bubble analogy in more detail. Imagine a tank of water in which we may have pellets of say plastic foam (electron analogue) and pellets of say metal (proton analogy). There is a gravitational potential 9.8 z increasing vertically which we take as the electrostatic potential analogue.

To work with hard numbers let's take the pellets to all have 1cc volume, recalling that water weighs 1gram per 1 cc. Let the foam pellets weigh 0.2 grams and the metal pellets weigh 1.8 grams.

All dynamic forces will try to minimize the potential energy. Note that lifting a positive mass increases its gravitational potential and its potential energy. But since we are dealing with balls in water lifting a ball also involves water moving out of the volume the ball will occupy and into the volume the ball formerly occupied. As a ball rises effectively an equivalent volume of water lowers by the same amount.

Imagine a metal pellet held at the top of the 1 meter water column. Imagine you freeze time so the water will not flow until you restart the clock. Snap your fingers and make the metal ball disappear. What is left is a spherical void in the water. You must fill it. So at the bottom of the column you snap your fingers and make an identical sphere of water teleport up to fill the void. You finally snap your fingers a third time and the metal ball reappears in the lower void.

The net effect is the metal ball (1.8gram) has moved downward 1 meter. The water ball (1gram) moved upward 1 meter. The net change in potential energy is as if you moved a 0.8 gram mass downward 1 meter. Got that?

The change in potential energy is -1m x 0.8/1000 kg x 9.8 m/s^2 = -0.00784 J.
You can think of the metal ball as having an effective mass of only 0.5 grams (its mass minus the mass of the water it displaces.)

Now do the same with a foam ball and since it masses 0.8 grams less than the water it has a negative effective mass of -0.8 grams. Lowering the foam ball as with the metal ball yields a change of potential energy of (-1m)x (-0.8/1000 kg) x 9.8m/s^2 = +0.00784J.

In order for potential energy to decrease the foam ball must rise (to let the denser water fall). This is Archimedes principle of bouyency.

I use it here as an analogue to electrical charges in a potential. The positive charges "sink" to the lower electrical potential just as the metal balls sink to lower gravitational potential. Negative charges "float" to higher electrical potential just as the foam balls float to higher gravitational potential. In both cases the potential energy is lowered.

Again the water is there just to get the two types of balls to behave in opposite fashon to preserve the analogy with electric charges. We don't need to invoke some sea of positive charge which negatively charged electrons displace in order to "move backward" but you might consider it temporarily to get a sense of the interactions of negative and positive signed charges with the potential.

Remember both the signs we assign to charges and the signs we assign to the electrical potentials they generate are both in a sense arbitrary. If as I suggested we flip everything to let electrons have positive e charge then the field produced by a clump of protons will also reverse signs and so the direction of motion will still be so that electrons are pulled toward the protons et vis versa.

Start with that in any problem. Which way would a proton be pulled? That's "down-hill" to lower electrical potential and that's the direction of the E field. Since proton's repel protons the field generated by another proton must decrease as you move away and the E field points outward. The field generated by electrons are opposite and the behavior of electrons is opposite. These two opposites cancel so electrons repell electrons as well.

BTW Introducing a "sea" of charge has been used for other reasons (look up: Dirac Sea and anti-matter) and is approprite in describing semi-conductors where you have holes in the Fermi sea of electrons acting like positively charged particles.