Conducting Circles Conducting Parallel Bars 1. In theory, the electric field sho
ID: 1838915 • Letter: C
Question
Conducting Circles Conducting Parallel Bars
1. In theory, the electric field should be constant along the axis of symmetry for the parallel conductin bars. Find the mean of the calculated E values form the data table and the standard error. report the final experimental value for the electric field as (mean + - SE)
3. Where is the magnitude of the electric field the largest and the smallest for the Conducting Circles?
Va(V) Vb(V) d(m) E(V/m) Va(V) Vb(V) d(m) E(V/m) 10.00 8.00 0.015 133.33 10.00 8.00 0.02 100 8.00 6.00 0.022 90.91 8.00 6.00 0.023 86.96 6.00 4.00 0.038 52.63 6.00 4.00 0.025 55.56 4.00 2.00 0.026 76.92 4.00 2.00 0.022 90.91 2.00 0.00 0.014 142.86 2.00 0.00 0.026 76.92Explanation / Answer
Part 1: Mean E for parallel conducting lines:
Since the values are already given(calculated by experiment), calculating the mean of the values is done simply by taking sum of the values and then divide it by the number of observations. In our case (for conducting lines) the values are:-
Sum = 100 + 86.96 + 55.56 + 90.91 + 76.92 = 410.35
Number of observations = 5
Since for parallel conducting lines the potential essentially remains the same there is no concept of weighted average in this case, hence the mean is simply 410.35 / 5 = 82.07
For calculating the standard error there are actually two ways. Typically for an experiment the method is to check values from both the methods and see whether they are within an acceptable error range.
Method 1: dr(E)/E = dr(Va)/Va + dr(Vb)/Vb + dr(d)/d (read dr as derivative). Essentially here we are assuming that experimental error can happen from all the variables that are used in the formula E = (Va - Vb) / d. So we calculate the maximum error that can happen keeping in mind of all the variables in the equation. Now as far as the values of the each terms are concerned those basically turn out to be approximately standard deviations or the errors associated with each term. Hence the sum of the standard deviations of the all 3 terms = max error that can happen to your E.
For our experiment that comes out as 3.16, 3.16, 0.0023 respectively. Summing them up we get dr(E)/E = 6.32
Methods 2: Although for physical experiments this is by far the better method relative to the previous one, but since the number of observation in our case is low, this might give more error than expected in this case. Standard error statistically is defined as (Standard Dev of Observations) / (square root of (number of observations)). Although statistically this is sound, but for physical process (or experiments in our case) here we are not accounting for the errors that can happen due to the other terms in the formulae. But by applying this method we obtain the result as follows:
16.976 ~ 17 / sqrt(5) = 7.59 ~ 7.6 (~ means approximately)
So from the two methods we get the results as 6.32 and 7.6 which is relatively close to each other considering the range of the electric field values. If you get confused as to which one to use just take an average of these two values and use that. In our case that comes out as 6.96 ~ 7. So the Electric field can be written as E = 82.07 + 7
Part 2: Imagine a single point in space. Lets name the point X. For the time being lets assume that there exist intersecting equipotential lines. Now lets have potential source points A and B which are of different potential energies but are equidistant from X. We are going to calculate the value of electric field between A and X and then between B and X. So going by the equation:
E1 = V(A) - V(X) / d
E2 = V(B) - V(X) / d
Since V(A) and V(B) are not equal, E1 and E2 are not equal as well. Hence X exist at different potential simultaneously. That is not physically possible. Hence our initial assumption is incorrect. Hence the lines from A->X and B->X cannot exist simultaneously. Hence there is no possibility that they can intersect.
Part 3: For conducting circles or rings (I'm assuming that we are dealing with 2D objects here) there exist electric field only in the direction perpendicular to the plane of the ring. Which means that suppose you have a ring lying on the X-Y plane, then the electric field will be in the Z direction. The value of the electric field is inversely proportional to the distance of the point of measurment from the center of the ring. Which means that the value of the field is maximum at the center of the ring and decreases as you move away from it. Hence Largest will be at the center of the ring and smallest will be at infinity (theoretically).