Consider a capacitor made of two square plates of side l. The distance between the two plates is d .

(a) We insert a dielectric of dielectric constant K>1 and width l a distance x (as in the diagram). What is the total capacitance of this arrangement? Express your answer in terms of the following variables and constants, if relevantl, x , d , K , and epsilon_0

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(b) The capacitor is now connected to battery which provides a difference of potential Vo across the capacitor. What is the energy stored in the capacitor?

Express your answer in terms of the following variables, if relevantl, x , ,d , K , Vo and epsilon_0

(c) While the battery is still connected to the capacitor, we now move the dielectric slab a bit further in between the plates, increasing x by an amount Δ. What is the change in the energy stored in the capacitor?

Express your answer in terms of the following variables, if relevantl,x , d , K ,Vo, Δ , and epsilon_0

What is the work done by the battery while we push the dielectric slab in from its original position x to x+Δ ? (make sure you have the correct sign!) Express your answer in terms of the following variables, if relevant l, x ,d , K ,Vo ,Δ and epsilon_0

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What is the work done by us while we push the dielectric slab in? (make sure you have the correct sign!) Express your answer in terms of the following variables, if relevant,l , x ,d , K, Vo, Δ and epsilon_0

Please someone answer.

While the battery is still connected to the capacitor, we now move the dielectric slab a bit further in between the plates, increasing x by an amount Δ. What is the change in the energy stored in the capacitor? What is the work done by the battery while we push the dielectric slab in from its original position x to x+Δ? What is the work done by us while we push the dielectric slab in? Make sure you have the correct signs!

(a) To calculate the total capacitance of the arrangement, we can use the formula for the capacitance of a parallel plate capacitor with a dielectric:

C = (K * epsilon_0 * A) / d

Where:
C = Total capacitance
K = Dielectric constant
epsilon_0 = Permittivity of free space
A = Area of one plate
d = Distance between the plates

In this case, the area of each plate is l^2, and the distance between the plates is d. Therefore, the total capacitance is:

C = (K * epsilon_0 * l^2) / d

(b) The energy stored in a capacitor can be given by the formula:

U = (1/2) * C * V^2

Where:
U = Energy stored
C = Total capacitance
V = Potential difference across the capacitor (Vo in this case)

Substituting the value of C from part (a), we get:

U = (1/2) * ((K * epsilon_0 * l^2) / d) * Vo^2

(c) When the dielectric slab is moved further between the plates, the capacitance of the arrangement increases. The change in energy stored can be calculated by taking the difference between the final energy and the initial energy:

ΔU = U_final - U_initial

Using the formulas from part (b), and considering the change in capacitance due to the change in x:

ΔU = (1/2) * ((K * epsilon_0 * l^2) / (d + Δ)) * Vo^2 - (1/2) * ((K * epsilon_0 * l^2) / d) * Vo^2

The work done by the battery while pushing the dielectric slab in from x to x+Δ is equal to the change in energy stored in the capacitor:

Work_battery = ΔU

The work done by us while pushing the dielectric slab in is equal to the negative of the work done by the battery:

Work_us = -ΔU