Electrostatics Intro
Happy New Year!
I wanted to bring up a point which I've noticed mentioned a lot on this blog here: http://jacobsphysics.blogspot.co.uk/
What point is that exactly? Well, I think it's that vectors have a magnitude and a direction. Yep. That definition which you memorised in Year 8 or maybe Year 9 (or for GCSE, or As level) is well... correct. And very helpful. Because a lot of physics confusion arises from the difficult issue of signs- whether something is negative or positive. But here's the thing: negatives are not real. They work, great, but have you ever seen a 'negative' apple? Have you ever picked up a weight that weighs -4kg? No, you haven't. Because those things don't exist.
Let me explain, even though this seems obvious (hopefully). When we do kinematics, it's often helpful to asign a direction as positive and another direction as negative. The maths works this way, so we do it. And that's fine and all, but what you mean by g=-9.81N/m is not a negative value, but a value that is opposite to your positive direction. This confusion doesn't get any clearer when you hear people explaining that something is a vector quantitiy because it can be positive or negative (like acceleration, right? Or velocity? These things can totally be negative...)
Vectors are vectors because they have magnitude and direction. It might be helpful in a 1-D situation to show that direction by a +ve or a -ve sign, but that is not enough to make it a vector! In fact, if you had a quantity that could only act in a line, and only forwards and back (i.e. it always is directed 'up' or 'down', but never sideways) then it probably isn't a vector. It probably doesn't actually exist for that matter. Velocity can be pointed in any direction at all, not just a crude positive or negative. And the nail in the coffin is electric charge, because charges can be negative, so they must be vectors! Of course not, that doesn't make any sense! What direction does a negative charge face? That's not a well-defined question and it cannot have a sensible answer. Charges are scalars, by the way. The idea of positive and negative means something completely different to direction and yet is ingenius.
The post was titled 'an intro to electrostatics' so I'd better actually talk about that a little.
Fact #1: Like charges attract, and unlike charges repel
This is what leads to the (very clever) use of signs. Because the above fact suggests that there are two types of charge. More importantly, the type of charge doesn't really matter. What matters is whether they are the same or not. This actually resemles a mathematical idea: the product of two numbers is positive if they are both negative or if they are both positive. The product is negative if they have different signs. This similarity means that you can represent the different types of charge as scalars with different signs and use ordinary algebra to represent what's going on! (Force depends on the product of two charges, this means that when the product is +ve, you get repulsion and when it's -ve you get attraction. Again, the sign indicates a direction as force is a vector, where +ve is loosely taken to be "away from the centre" and -ve therefore is "towards each other").
Fact #2:electric fields exert forces on charges
This is the main fact. The big one. The F=ma of electrostatics. Expcept in this case it's F=qE.
An electric field is something that exerts a force on a charge, and is defined using that simple equation. Now, electric fields have some vital properties you need to know The first is so importnt it needs to be a seperate fact:
Fact #3: positive charges have a force in the direction of an electric field, negative charges have the opposite direction.
Why is this? Well, the actual bit of physics is that unlike charges experience opposite forces in the same field. The bit about the direction being the same is simply because the direction of an electric field is defined to be the direction it accelerates a positive 'test' charge. The final fact about electric fields is less interesting to be honest: you can add them up. Fields are vectors, so they add using vector arithmetic (triangles and cosines I'm afraid). The force is then found using F=qE, where E is the total field.
In summary: you have a bunch of charges and a test charge. You want to know the force on the test charge. So... find the field from each charge at the position of your test charge, then add them together and multiply by the magnitude of the test charge. If the test charge is positive, the direction of the force is the same as your resultant field, it it's negative then it's the opposite direction. So simple, so easy: just work out a bunch of fields, add them up, and multiply by a charge.
