Question

# How can we change flux through a given area?

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Solution

## Step 1: Definition of FluxConsider a surface element $d\stackrel{\to }{S}=\stackrel{^}{n}dS$ in an electric field $\stackrel{\to }{E}$, where $\stackrel{^}{n}$ is the outward unit vector normal to the surface element.The quantity $d\varphi =\stackrel{\to }{E}.d\stackrel{\to }{S}$ becomes equal to the number of lines of force passing through the area $d\stackrel{\to }{S}$.The flux of $\stackrel{\to }{E}$ over any arbitrary surface $S$ is given by the integral, ${\varphi }_{E}=\int d\varphi ={\int }_{S}\stackrel{\to }{E}.d\stackrel{\to }{S}$Step 2: Magnetic FluxFor magnetic flux, if $S$ is an open surface bounded by the curve $C$ placed in a magnetic field $\stackrel{\to }{B}$ then the magnetic flux through the surface is,${\varphi }_{B}={\oint }_{S}\stackrel{\to }{B}.d\stackrel{\to }{S}$Step 2: Flux versus AreaTherefore from the above formula, we can see that flux varies with the area $d\stackrel{\to }{S}$.Hence, for a constant electric field, the flux passing through the surface element is directly proportional to the area $d\stackrel{\to }{S}$.In order to change flux by keeping the area constant either $\stackrel{\to }{B}$, $\stackrel{\to }{E}$ or $\theta$ must be changed.Hence, flux changes proportionally with the area.

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