derive the formula for electric field due to a dielectric
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Dielectric constant KK is actually the same thing as relative permittivity, and it increases the overall permittivity ϵϵ. So in general, whenever you see the permittivity of free space ϵ0ϵ0 in an equation, if you're dealing with a dielectric, you can multiply it by the dielectric constant and see how the equation changes.
For example, since C=ϵ0ADC=ϵ0AD, multiplying ϵ0ϵ0 by KK increases capacitance by KK. Since ∮E⋅dA=Qϵ0∮E⋅dA=Qϵ0 by Gauss's Law, multiplying ϵ0ϵ0 by KK decreases electric field magnitude by KK.
There is no contradiction because there is no increase in electric flux. The dielectric decreases the electric field magnitude, which decreases the electric flux and decreases the voltage across a capacitor as well. C=QVC=QV and the capacitance increases because the voltage decreases while the charge remains the same.
For example, since C=ϵ0ADC=ϵ0AD, multiplying ϵ0ϵ0 by KK increases capacitance by KK. Since ∮E⋅dA=Qϵ0∮E⋅dA=Qϵ0 by Gauss's Law, multiplying ϵ0ϵ0 by KK decreases electric field magnitude by KK.
There is no contradiction because there is no increase in electric flux. The dielectric decreases the electric field magnitude, which decreases the electric flux and decreases the voltage across a capacitor as well. C=QVC=QV and the capacitance increases because the voltage decreases while the charge remains the same.
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