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Two equal sized charged spheres, sphere π΄ and sphere π΅, have charges of plus π one and negative π two respectively.
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The spheres are separated by a straight-line distance π.
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The net electric field produced by the spheres has a null point along π at a distance of π over four meters from sphere π΄.
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What is the ratio of the magnitude of the charge of sphere π΅ to that of sphere π΄?
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Letβs start off by drawing a diagram of these two spheres.
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Our scenario involves two charged spheres, sphere π΄ with charge positive π one and sphere π΅ with charge negative π two.
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They are separated by a distance π.
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And weβre further told that at a point on this line a distance of π over four meters from sphere π΄, the electric field is zero.
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The statement asked us to solve for the ratio of the magnitude of the charges π two and π one.
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We can start figuring that out by recalling the equation for the electric field created by a point charge.
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A point charge, π, creates an electric field, πΈ, equal to the magnitude of the charge times π, Coulombβs constant, divided by π squared, where π is the distance from the charge to the point where the field πΈ is being measured.
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Our diagram shows that the charges on sphere π΄ and π΅ create an electric field of zero at a particular location.
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If we call that field capital πΈ, then πΈ equals π times π one, the charge on sphere π΄, divided by π over four quantity squared plus π times π two, the charge on sphere π΅, divided by the distance three π over four quantity squared.
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This sum, weβre told, is equal to zero.
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Considering this equation, we see that we can cancel out both the factors of π and the factors of π over four in our denominators.
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That leaves us with an equation saying that π one plus π two over three squared, or nine, is equal to zero or that π two equals negative nine π one.
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Since our problem statement specifically asked for the magnitude of the ratio of π two to π one, we can remove the minus sign we see here.
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And we see that the magnitude of charge π two equals nine times the magnitude of charge π one.
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Thatβs the charge magnitude ratio that creates this electric field null point.