(II) How much work would be required to remove a metal sheet from between the plates of a capaci

(II) How much work would be required to remove a metal sheet...

Key Concepts

Work Done in Changing Capacitor Geometry

The work done in processes such as removing a metal sheet from between capacitor plates comes from the change in the stored electrostatic energy. This work can be determined by calculating the difference in energy before and after the modification, using the appropriate energy expressions for constant voltage or constant charge conditions. Understanding this energy difference is crucial to predicting the mechanical effort required in practical scenarios.

Energy Stored in a Capacitor

This concept involves the calculation of energy stored in a capacitor, which is central to understanding the work done when modifying capacitor configurations. The stored energy can be expressed as one-half of the product of capacitance and the square of the voltage (½CV²) when the capacitor is connected to a battery, or in terms of charge and capacitance (Q²/2C) when the charge is kept constant. Understanding how these formulas change when the capacitor’s geometry or configuration is altered is vital in determining the work required for such processes.

Capacitance and Its Dependence on Geometry

Capacitance is determined by the physical characteristics and arrangement of the capacitor, including the area of the plates and the separation between them. When a metal sheet is introduced or removed, it effectively alters these geometrical factors, which in turn changes the capacitance of the system. Grasping how the capacitance is affected by such modifications is key to analyzing the changes in stored energy and the associated work done.

Constant Voltage Versus Constant Charge Conditions

The difference between having a constant voltage and a constant charge is fundamental to understanding energy variations in capacitors when their configurations change. Under constant voltage (battery connected), the voltage remains fixed and the capacitance change directly influences the stored energy. Under constant charge (battery disconnected), the total charge remains unchanged, and any change in capacitance inversely affects the potential difference and energy. Recognizing these differences is essential in correctly determining the work required for altering the capacitor’s state.

Transcript

00:01 To solve this question, we need to find the capacitance of the capacitor, which is somewhat like this.

00:10 This has thickness of l.

00:12 So capacitance here, this one, and this one, they are same.

00:19 There are two capacitors in series.

00:22 So the capacitor, capacitance here is given by twice a epsilon not divided by d.

00:32 Minus l since this distance is d minus l divided by same is the case with this so the net capacitance of these two capacitors is cn equals 1 over c plus 1 over c and if you solve it you get c n equals a epsilon not divided by d minus l let's also calculate the charge stored on the capacitor when it is connected to the battery q equals cv, or just cv is enough.

01:28 So that will be a epsilon not v divided by d minus l.

01:43 Now to solve the first part which says find the change in energy when the plate, is removed while the battery is connected.

02:00 That means the potential is the same.

02:02 So in this case, you can simply use dw is equal to half.

02:09 Since when you remove the plate, you're changing the capacitance.

02:12 So dc, v square.

02:17 So let's find dc.

02:18 Dc is initial capacitance minus the final capacitance.

02:28 From this, this is your initial capacitance.

02:32 Initial capacitance now is a epsilon not divided by d minus l final capacitance is a epsilon not divided by d since you remove the plate now this gives you a epsilon not l divided by l d minus l i'm sorry this should be d now, just substitute that to find the work done.

03:17 So the work done, in this case, is equal to v squared divided by 2, a epsilon 0, l, i'm sorry, again, d, d minus l.

03:40 Now, in the second part, we are supposed to consider that battery is disconnected.

03:47 That means the charge is constant...

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