From Graphical Analysis

Question

Please help me with this. It looks long, but it really is not.

A physics student wants to determine the relationship between the radius of a charged conducting sphere and the magnitude of the surface charge. The student suspects a relation of
V=(kq)/(r^n)
Assume that the voltage is held constant at 10V, and that k is a constant of 9*10^9 
F/m. The student uses different spheres of radius (r)=(5cm, 10cm, 15cm, 20cm, 25cm). The respective charge values at q=(55pC, 111pC, 165pC, 220pC, 280pC)
From this data, create a plot to determine the value of n.
what is the value of n?
What is the correlation of coefficient of the plot?

Answer 1

Hey Michaela -- in this data set, q/r is a constant [eyeballing a factor of "11"] ...

 

the exponent "n" in this case must be "1" [n=1] ... the correlation is super-strong ...

 

only the 280pC over 25cm is a bit off the "11" at "11.2" ... Very best wishes, ma'am :)

Answer 2

First solve the relationship V=kq/rⁿ for the variable you are measuring, q:

q = (V/k) rⁿ.

You want to find the exponent n graphically from your data. Without graphing software, the best way to do this is to compute the natural logarithms of all your q- and r-data, and then graph ln(q) vs. ln(r). This will turn the power function q into a linear function, ln(q):

 

ln(q) = ln(V/k) + n ln(r).

 

When you compare this with the equation of a straight line, y=mx+b, you see that the slope of this graph will be the exponent n you are looking for. (The term ln(V/k) is the y-intercept.)

 

Alternatively, you can graph q vs. r on a special kind of graph paper called doubly logarithmic paper, which will also turn a power function into a straight line.