Electric Potential & U _{E} Lesson 02 |

Hey everyone... power's out here in Kitsap. I have my generator fired up and I'm camped out around my space heater. I do have limited internet access through my phone, but I can't do Hangouts today. Please review the following lesson plan... the partial derivative stuff may be kinda fun, but that's more of an aside than anything we have to actually get to (if you look in the back of your calc book you'll see calculus with multiple derivatives and integrals <hence the term multi-variant calculus>. That is usually taught as 3rd semester calc.) Finally -- I can't emphasize how important it is to keep up with the work. We are NOT going to spend ANY time catching up whenever it is that we return to class. Also, there's a very good chance we'll have a test THIS week if we ever make it back... so stay frosty (so to speak)
OBJECTIVES: 1) I will be able to calculate electrical potential by analyzing the contribution of a few charged particles after today's class. 2) I will be able to calculate electrical potential by analyzing a continuous series of charged particles (using integration) after today's class. 3) I'm going to get my first real look at partial derivatives and how to apply them in tonight's homework.
WORDS/FORMULAE FOR TODAY Terms:
Constants:
Formulae: - Electric Potential (V) and Electric Potential Energy (U
_{E})- ∆U = -q∫E ∙ ds
- V
_{b}- V_{a}= ∆V = ∆U/q = -∫E ∙ ds - W = q∆V = -U
_{E} - V = k
_{e}∑q_{i}/r_{i}(between some point in space and each charged particle present in that region of space) - U
_{E}= k_{e}∑q_{1}q_{2}/r_{i}(between each particle in space and every other particle in that region of space)
- Electrical Force:
- F
_{e}= (k_{e}q_{1}q_{2})/r^{2} - F
_{e}=qE
- F
- Electrical Field:
- E = F
_{e}/ q_{o}where q_{o}= a positive test charge - E= (k
_{e}q)/r^{2}
- E = F
- Gauss's Law:
Φ = ∮E ∙ dA = ∮Ecos ∙ dA = q _{in}/ε_{o}Φ = q _{in}/ε_{o}: the electric flux through an ENTIRE gaussian surface is equal to the algebraic sum of the charges INSIDE the surface divided by the permitivity of free space
WORK O' THE DAY: Spend a few moments familiarizing yourself with the relationships between ∆U, ∆V and such...this needs to be right on the tip of your tongue... NOW! ════════════════════ There are two important considerations in section 25.3:
════════════════════ (FUN WITH NUMBERS!!!! (hold on to your hats folks! We *may* get to this at the VERY end of the term, but for now I'd like you to take a first look at it-- please do NOT stress over this) Recollect (?) that: V since we often replace ∆ with 'd' when things are getting dV = -∫E ∙ ds let's simplify that further for the case of a uniform electric field: dV = -E∫∙ ds or even more simply: dV/ds = -E Since s is a generic distance term, let's rewrite that using our generic 'r' for displacement: dV/dr = -E however, if we consider that the 'r' vector is often rewritten in terms of the x, y and z vectors we can rewrite that as the vector sum of the -E = ∂V/∂x + ∂V/∂dy + ∂V/dz so...let's say we can describe the electric potential due to an object by the equation: E(r) = 2x + 3xy + 4z E(x,y,z) = 2x + 3xy + 4z so it follows that: ∂V/∂x = 2+3y ∂V/∂dy = 3x ∂V/dz = 4
Now... back to the task at hand-- If we are considering only the x direction we have just a piece of that: -E = ∂V/∂x BUT WAIT... recall that F -Eq = q∂V/∂x HOLD THE BUS!!!! qV = U and E -F which is an awful lot like: our old friend: -F ════════════════════ We can't *always* deal with the electric potential from a countable number of charged particles. In otherwords, we sometimes need to deal with a continuous number of charged particles.... and that means calculus! We often begin with the VERY small: We can evaluate a tiny, tiny, tiny bit of electric potential: dV = (K And then use integration to sum up that tiny bit of electric potential over MANY charges: ∫dV = ∫(K which goes to: V = K In order to make THAT particular fish fry, we will need to get q in terms of r... let's take a look at our favorite long, thin wire ( HOMEWORK: 1) Review example 2) WORK example 3) Homework problem 4) Homework problem |

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