Figure shows two identical parallel plate capacitors connected to
Question: Figure shows two identical parallel plate capacitors connected to a battery through a switch S. Initially, the switch is closed so that the capacitors are completely charged. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of dielectric constant 3 . Find the ratio of the initial total energy stored in the capacitors to the final total energy stored. Solution:...
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Question: Differentiate $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ Solution: $1-\sin \theta=\left(\cos \frac{\theta}{2}-\sin \frac{\theta}{2}\right)^{2}$...
Read More →A capacitor is formed by two square metal-plates
Question: A capacitor is formed by two square metal-plates of edge a, separated by a distance $d$. Dielectrics of dielectric constants $K_{1}$ and $K_{2}$ are filled in the gap as shown in figure. Find the capacitance. Solution:...
Read More →Find the capacitances of the capacitors shown in figure.
Question: Find the capacitances of the capacitors shown in figure. The plate area is A and the separation between the plates is d. Different dielectric slabs in a particular part of the figure are of the same thickness and the entire gap between the plates is filled with the dielectric slabs. Solution:...
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Question: Differentiate $\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ with respect to $\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$ Solution:...
Read More →A parallel-plate capacitor having plate area
Question: A parallel-plate capacitor having plate area $400 \mathrm{~cm}$-square and separation between the plates $\mathrm{I} .0 \mathrm{~mm}$ is connected to a power supply of $100 \mathrm{~V}$. A dielectric slab of thickness $0.5 \mathrm{~mm}$ and dielectric constant $5.0$ is inserted into the gap. (a) Find the increase in electrostatic energy, (b) If the power supply is now disconnected and the dielectric slab is taken out, find the further increase in energy, (c) Why does the energy increas...
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Question: Differentiate $\sin ^{3} \mathrm{x}$ with respect to $\cos ^{3} \mathrm{x}$. Solution:...
Read More →A parallel-plate capacitor has plate area
Question: A parallel-plate capacitor has plate area $100 \mathrm{~cm}$-square and piate separation I.0 cm. A glass plate (dielectric constant $6.0$ ) of thickness $6.0 \mathrm{~mm}$ and an ebonite plate (dielectric constant 4.0) are inserted one over the other to fill the space between the plates of the capacitor. Find the new capacitance. Solution:...
Read More →A parallel-plate capacitor of capacitance 5 μF is connected to a battery of emf 6 V.
Question: A parallel-plate capacitor of capacitance $5 \mu \mathrm{F}$ is connected to a battery of emf $6 \mathrm{~V}$. The separation between the plates is 2 $\mathrm{mm}$. (a) Find the charge on the positive plate; (b) Find the electric field between the plates, (c) A dielectric slab of thickness 1 $\mathrm{mm}$ and dielectric constant 5 is inserted into the gap to occupy the lower half of it. Find the capacitance of the new combination, (d) How much charge has flown through the battery after...
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Question: Differentiate $\tan ^{-1}\left(\frac{\mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}}\right)$ with respect to $\cos ^{-1}\left(2 \mathrm{x}^{2}-1\right)$ Solution:...
Read More →A capacitor stores 50 μC charges when connected across a battery.
Question: A capacitor stores $50 \mu \mathrm{C}$ charges when connected across a battery. When the gap between the plates is filled with a dielectric, a charge of $100 \mu \mathrm{C}$ flows through the battery. Find the dielectric constant of the material inserted. Solution:...
Read More →The separation between the plates of a parallel-plate
Question: The separation between the plates of a parallel-plate capacitor is $0.500 \mathrm{~cm}$ and its plate area is $100 \mathrm{~cm}-\mathrm{square}$. A $0.400 \mathrm{~cm}$ thick metal plate is inserted into the gap with its faces parallel to the plates. Show that the capacitance of the assembly is independent of the position of the metal plate within the gap and find its value. Solution:...
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Question: Differentiate $\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$ with respect to $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ Solution:...
Read More →If the above capacitor is connected across
Question: If the above capacitor is connected across a $6.0 \mathrm{~V}$ battery, find (a) the charge supplied by the battery, (b) the induced charge on the dielectric and (c) the net charge appearing on one of the coated surfaces. Solution:...
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Question: Differentiate $\tan ^{-1} \sqrt{\frac{1-\mathrm{x}^{2}}{1+\mathrm{x}^{2}}}$ with respect to $\cos ^{-1} \mathrm{x}^{2}$ Solution:...
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Question: Differentiate $e^{\sin x}$ with respect to $\cos X$. Solution:...
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Question: Differentiate $\log \mathrm{X}$ with respect to $\cot \mathrm{X}$. Solution:...
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Question: Differentiate $\mathrm{X}^{6}$ with respect to $(1 / \sqrt{\mathrm{x}})$ Solution:...
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Question: If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\ldots \infty}}}$, show that $(2 y-1) \cdot \frac{d y}{d x}=\frac{1}{x}$. Solution:...
Read More →Consider the situation shown in figure.
Question: Consider the situation shown in figure. The switch $S$ is open for a long time and then closed, (a) Find the charge flown through the battery when the switch $\mathrm{S}$ is closed, (b) Find the work done by the battery. Solution:...
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Question: If $\mathrm{y}=\sqrt{\tan \mathrm{x}+\sqrt{\tan \mathrm{x}+\sqrt{\tan \mathrm{x}+\ldots \infty}}}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sec ^{2} \mathrm{x}}{(2 \mathrm{y}-1)}$ Solution:...
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Question: If $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x}+\ldots \infty}}$, prove that $\frac{d y}{d x}=\frac{\sin x}{(1-2 y)}$ Solution:...
Read More →A capacitor having a capacitance of 100 μF is charged to
Question: A capacitor having a capacitance of $100 \mu \mathrm{F}$ is charged to a potential difference of $24 \mathrm{~V}$. The charging battery is disconnected and the capacitor is connected to another battery of emf $12 \mathrm{~V}$ with the positive plate of the capacitor joined with the positive terminal of the battery, (a) Find the charges on the capacitor before and after the reconnection. (b) Find the charge flown through the $12 \mathrm{~V}$ batteries, (c) Is work done by the battery or...
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