Capacitor With A Dielectric

When the wire is stretched, its length is halved. The resistance of a wire is given by: where is the resistivity, is the length, and is the cross-sectional area of the wire. - When the wire is stretched, the volume remains constant, so , where the subscript 1 and 2 represent the initial and final conditions. The final resistance is given by: Since the length is halved, the final resistance will be:

Given:
- Wavelength
- Work function
- Planck's constant
- Speed of light The energy of the incoming photon is: The kinetic energy of the emitted photoelectron is: The maximum velocity of the photoelectron is given by: The linear momentum of the photoelectron is:

Capacitance of a Parallel Plate Capacitor with Partial Dielectric Slab:
Consider a parallel plate capacitor with plate area and separation between the plates. Let the capacitor be partially filled with a dielectric material of dielectric constant and thickness . The remaining gap is filled with air. The total capacitance can be considered as the combination of two capacitors in series: 1. The capacitor with dielectric slab: Capacitance , 2. The capacitor with air gap: Capacitance . 1. Capacitance with dielectric slab: The capacitance of the part filled with the dielectric is given by: 2. Capacitance with air gap: The capacitance of the part filled with air is given by: 3. Total Capacitance (Series Combination): Since the two capacitors are in series, the total capacitance is given by: Substituting the expressions for and : Simplifying the expression: Thus, the total capacitance is: So, the capacitance of a parallel plate capacitor with a partial dielectric filling is the same as that of an empty capacitor, but with the dielectric slab affecting the effective area.

Capacitance of a Parallel Plate Capacitor: A parallel plate capacitor consists of two conducting plates separated by a distance , with area of each plate. The capacitance of such a capacitor is given by the formula: where: - is the capacitance,
- is the permittivity of free space,
- is the area of one of the plates,
- is the separation between the plates.
Effect of Dielectric Medium: When a dielectric medium with dielectric constant (also known as relative permittivity) partially fills the space between the plates, the capacitance is modified. The dielectric constant increases the capacitance by a factor of . If the dielectric only fills part of the space between the plates, the system can be treated as two capacitors in parallel: 1. One part of the capacitor (with area ) is filled with the dielectric. 2. The other part (with area ) is filled with air or some other dielectric with permittivity .
Capacitance of the Region with Dielectric: The capacitance for the part of the capacitor with dielectric is: where: - is the area of the plates covered by the dielectric,
- is the distance between the plates covered by the dielectric.
Capacitance of the Region with Air: The capacitance for the part of the capacitor with air (or vacuum) is: where:
- is the area of the plates not covered by the dielectric,
- is the distance between the plates not covered by the dielectric.
Total Capacitance: Since the areas and add up to the total area , and the distances and add up to the total distance , the total capacitance is the sum of the capacitances of the two regions: Formula for Capacitance: For the case where the dielectric partially fills the space between the plates (i.e., ), the final formula for the capacitance becomes: This is the general formula for the capacitance of a parallel plate capacitor partially filled with a dielectric medium.