# Scalar and Vector Quantities

1. Aug 31, 2009

### 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.

2. Sep 1, 2009

### 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.

3. Sep 1, 2009

### Red_CCF

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.

4. Sep 1, 2009

### Fightfish

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.

5. Sep 1, 2009

### Hootenanny

Staff Emeritus
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

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

6. Sep 1, 2009

### Red_CCF

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?

7. Sep 1, 2009

### dx

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.

8. Sep 1, 2009

### jambaugh

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.

9. Sep 1, 2009

### dx

The current through a surface is defined as ∫S <j,dA>. How can that be a vector?

10. Sep 1, 2009

### Red_CCF

But I thought that scalar only meant the magnitude, so positive only???

11. Sep 1, 2009

### dx

No, the standard meaning of the word 'scalar' is 'real number'. The magnitude is usually called 'absolute value'.

12. Sep 1, 2009

### Red_CCF

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

13. Sep 1, 2009

### Andy Resnick

Charge is a 3-form (See, for example, MTW, Gravitation, p113-4), so current density is a vector.

14. Sep 1, 2009

### Hootenanny

Staff Emeritus
Energy and time can both be negative.
I never said current density wasn't a vector.

15. Sep 1, 2009

### Red_CCF

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.

16. Sep 1, 2009

### jambaugh

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.

17. Sep 1, 2009

### jambaugh

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.

18. Sep 1, 2009

### Red_CCF

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?

19. Sep 1, 2009

### jambaugh

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.

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.

20. Sep 1, 2009

### Red_CCF

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.

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