An air condenser has an area of 100 cm^2 the plates being 4mm apart. It is connected in series with another condenser having plate area of 20cm^2 plates being separated by dielectric 0.1mm thick with E=2.5. If the potential difference across an condenser is 1000volts; find the potential difference across the combination.
To find the potential difference across the combination of the two capacitors connected in series, we need to use the formula for the total capacitance in a series combination:
1/C_total = 1/C1 + 1/C2
First, let's find the capacitance of each condenser individually:
For the air condenser (C1):
The area of the plates is given as 100 cm^2, which we can convert to m^2 by dividing by 10,000:
Area (A1) = 100 cm^2 = 100/10,000 m^2 = 0.01 m^2
The distance between the plates (d1) is given as 4 mm, which we can convert to m by dividing by 1000:
Distance (d1) = 4 mm = 4/1000 m = 0.004 m
The capacitance (C1) of the air condenser can be calculated using the formula:
C1 = (ε0 * A1) / d1
Where ε0 is the permittivity of free space, which is approximately 8.854 x 10^-12 F/m.
Plugging in the values:
C1 = (8.854 x 10^-12 F/m * 0.01 m^2) / 0.004 m
C1 = 22.135 x 10^-12 F
For the condenser with the dielectric (C2):
The area of the plates is given as 20 cm^2, which we can convert to m^2 by dividing by 10,000:
Area (A2) = 20 cm^2 = 20/10,000 m^2 = 0.002 m^2
The thickness of the dielectric (d2) is given as 0.1 mm, which we can convert to m by dividing by 1000:
Thickness (d2) = 0.1 mm = 0.1/1000 m = 0.0001 m
The relative permittivity of the dielectric (E) is given as 2.5.
The capacitance (C2) of the condenser with the dielectric can be calculated using the formula:
C2 = (E * ε0 * A2) / d2
Plugging in the values:
C2 = (2.5 * 8.854 x 10^-12 F/m * 0.002 m^2) / 0.0001 m
C2 = 44.27 x 10^-12 F
Now, we can find the total capacitance (C_total):
1/C_total = 1/C1 + 1/C2
1/C_total = 1/(22.135 x 10^-12 F) + 1/(44.27 x 10^-12 F)
1/C_total = (44.27 + 22.135)/(22.135 * 44.27 x 10^-24 F^2)
1/C_total = 66.405 / (981.244 x 10^-24 F^2)
1/C_total = 6.76 x 10^21 F^-2
C_total = 1 / (6.76 x 10^21 F^-2)
C_total = 1.48 x 10^-22 F
Finally, to find the potential difference across the combination, we can use the equation:
V_total = V1 + V2
Where V1 is the potential difference across the air condenser (1000 V given in the question) and V2 is the potential difference across the condenser with the dielectric.
Since the capacitors are connected in series, they have the same charge (Q). Therefore, the potential difference across each capacitor is proportional to its capacitance.
V2 = (C2 / C_total) * V_total
Plugging in the values:
V2 = (44.27 x 10^-12 F / 1.48 x 10^-22 F) * 1000 V
V2 = 2.99 x 10^10 V
Therefore, the potential difference across the combination of the two capacitors is approximately 2.99 x 10^10 volts.