Now we also know that the charge on a pair of parallel plates is proportional to the potential difference V, namely
q=CV.
We also know that the capacitance of two parallel plates is given by
C=ε0A/d,
where ε0=8.85×10-12 is the permittivity of free space and d is the separation between the plates. Combine these equations and to get the force as a function of the potential. So now if you were to make a graph of force (F) vs. V2/d2, the slope of the graph would be
A) E
B) ε0A/2
C) C
D) ε0
It's B
To combine the given equations and obtain the force as a function of potential, we can start by expressing capacitance (C) in terms of the permittivity of free space (ε0), plate area (A), and plate separation (d):
C = ε0A/d
Next, we substitute this expression for capacitance (C) into the charge equation:
q = CV
q = (ε0A/d)V
Now, we can rearrange this equation to express V in terms of q and the other variables:
V = (q / (ε0A/d))
V = (qd) / (ε0A)
Finally, we can express the force (F) in terms of q, d, and A using the equation for electric field (E):
F = Eq
F = (q / A) * E
F = (q / A) * (V / d)
F = (qV) / (dA)
Now, we can substitute the expression we derived for V into the above equation:
F = ((q * ((qd) / (ε0A))) / (dA))
Simplifying this expression further:
F = (q^2 * d) / (ε0A^2)
Now, we have the force (F) expressed in terms of q, d, and A. To find the slope of the graph of force (F) vs. V^2/d^2, we need to identify the coefficient in front of V^2/d^2 in the expression for force:
F = (q^2 * d) / (ε0A^2)
F = (q^2 / (ε0A^2)) * d
Comparing this expression to the desired form F = slope * (V^2 / d^2), we can see that the slope is:
slope = q^2 / (ε0A^2)
Therefore, the answer is D) ε0.