Math Problem Statement
The membrane of a living cell can be approximated by a parallel-plate capacitor with plates of area 4.85×10−9 m2 , a plate separation of 8.2×10−9 m, and a dielectric with a dielectric constant of 4.6.What is the energy stored in such a cell membrane if the potential difference across it is 7.55×10−2 V?
Solution
To find the energy stored in the parallel-plate capacitor, we can use the formula for the energy stored in a capacitor:
where:
- is the energy stored,
- is the capacitance of the capacitor,
- is the potential difference across the capacitor.
Step 1: Calculate the Capacitance
The capacitance of a parallel-plate capacitor is given by:
where:
- is the permittivity of free space, ,
- is the dielectric constant,
- is the area of the plates,
- is the separation between the plates.
Plugging in the given values:
Calculating :
Step 2: Calculate the Energy Stored
Now, using the formula for energy stored:
where .
Let’s calculate the energy stored.
I'll do the calculations for both capacitance and energy stored.### Calculated Results:
- Capacitance ():
- Energy Stored ():
The energy stored in the cell membrane, given the provided parameters, is approximately .
Would you like to know more details or have any questions?
Related Questions:
- How does the energy stored change if the plate separation is halved?
- What effect does doubling the dielectric constant have on the energy stored?
- How would the energy stored change if the potential difference is doubled?
- What is the physical significance of the dielectric constant in a capacitor?
- How can the concept of a parallel-plate capacitor be applied in practical biological systems?
Tip: In a capacitor, increasing the dielectric constant increases the capacitance, allowing more energy to be stored for the same voltage.
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Math Problem Analysis
Mathematical Concepts
Capacitance
Energy Stored in a Capacitor
Electric Potential
Formulas
U = 1/2 * C * V² (Energy stored in a capacitor)
C = (ε₀ * ε_r * A) / d (Capacitance of a parallel-plate capacitor)
Theorems
-
Suitable Grade Level
Undergraduate Physics
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