Given Two parallel plate capacitors with a seperation d(metres) in free space connected to a dc supply of volts,v. A dielectric material of thickness b(metres) is inserted between the plates so that a distance (d-b)m remains without dielectric. Show that the the pd per unit thickness is givenby v/e(d-b)+b where e is the dielectric constant.
To derive the expression for the potential difference per unit thickness, let's consider the two capacitors separately and then combine their results.
First, let's analyze the capacitor without the dielectric material. The capacitance of a parallel plate capacitor in free space is given by:
C1 = ε0 * A / d
Where ε0 is the permittivity of free space, A is the area of the plates in square meters, and d is the separation between the plates.
The charge on each plate of the capacitor without the dielectric can be determined using the formula:
Q1 = C1 * V
Where V is the applied voltage.
Now, let's analyze the capacitor with the dielectric material. The capacitance of a parallel plate capacitor with a dielectric material is given by:
C2 = (ε0 * e * A) / (d - b)
Where e is the dielectric constant.
Again, the charge on each plate of the capacitor with the dielectric can be determined using the formula:
Q2 = C2 * V
Now, let's consider the equivalent system where both capacitors are connected to the same voltage source. The total charge on each plate of the combined capacitor system will be the sum of the charges on each plate of the individual capacitors:
Q_total = Q1 + Q2
Now, let's substitute the expressions we derived earlier for Q1 and Q2:
Q_total = C1 * V + C2 * V
Q_total = (ε0 * A / d) * V + ((ε0 * e * A) / (d - b)) * V
Factoring out the common terms:
Q_total = (V / d) * (ε0 * A + (ε0 * e * A) * (d - b) / (d - b))
Simplifying further:
Q_total = (V / d) * (A * ε0 + A * ε0 * e * (d - b) / (d - b))
Q_total = (V / d) * (A * ε0 * (1 + e * (d - b) / (d - b)))
Q_total = (V / d) * (A * ε0 * (1 + e))
Since Q_total is the total charge on each plate of the combined capacitor, we can express it in terms of the equivalent capacitance C_total:
Q_total = C_total * V
Now, equating the two expressions for the total charge:
C_total * V = (V / d) * (A * ε0 * (1 + e))
Simplifying further:
C_total = (A * ε0 * (1 + e)) / d
Now, to find the potential difference per unit thickness, we divide both sides by the thickness of the dielectric (b):
(V / e(b - d)) = (A * ε0 * (1 + e)) / (d * b)
Rearranging the equation to isolate V / e(b - d):
V / e(b - d) = (A * ε0 * (1 + e)) / (d * b)
Thus, the potential difference per unit thickness is given by v/e(d - b) + b, where v is the applied voltage, e is the dielectric constant, and d and b are the separation and thickness of the dielectric, respectively.