<Text-field style="Heading 1" layout="Heading 1">Question 1</Text-field>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print(); # input placeholderLet LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRLFZlY3RvckZpZWxkRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiMtRlM2Ji1GIzYtLUklbXN1cEdGJDYlLUYsNiVRInhGJ0YvRjItRiM2Iy1JI21uR0YkNiRRIjJGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmhvLUZmbjYlLUYsNiVRInlGJ0YvRjJGW29GYW8tRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJ0Zdby1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRmNwL0ZORmRwRmhuRmdwRlxwRl9wLUZebzYkRmNvRjlGOS8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTQ== Calculate the electric potential function, assuming that the potential is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv at the origin.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Solutione</Text-field>The relation between the electric field and the electric potential is: 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 it means that we have to find a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSUYy that satisfies these equations: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 = print(); # input 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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiUtSSNtbkdGJDYkUSIwRidGMi1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGOkYyLUY/Ni1RIj1GJ0YyRkIvRkVGMUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGZm5GOi1GPzYtUSI6RidGMkZCRlpGR0ZJRktGTUZPRmVuRmduThe only function that satisfies all of those conditions is JSFHprint(); # input placeholder
JSFH
<Text-field style="Heading 1" layout="Heading 1">Question 2</Text-field>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A wire is laid as half a circle around the origin, on the positive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= direction. The wire is charged with a non-uniform line charge density 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 the angle is measured from the positive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis.Find the electric potential at the origin, assuming that the potential is zero at infinity. JSFH
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>We will calculate the potential of an infinitisimal charge, and sum it over the half circle:JSFHJSFHJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZFFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is the infinitisimal charge, per LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy increment. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoZCYjOTM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YvRjI= is the contribution of the infinitisimal charg to the total electric potential. 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print(); # input 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 = print(); # input placeholderSubstituting:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2I1EuYCZsYW1iZGE7X18wYEYnLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbW5HRiQ2JFEoMTEwLjYyNUYnRjMtSShtZmVuY2VkR0YkNigtRiM2JC1GLDYjUSFGJy1JJm1mcmFjR0YkNigtRiw2JVEscGljb2NvdWxvbWJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNFEnaXRhbGljRictRiw2JVEibUYnRmZuRmluLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zjby8lKWJldmVsbGVkR0Y4RjMvSSttc2VtYW50aWNzR0YkUSVVbml0RicvJSVvcGVuR1EoJmxvYnJrO0YnLyUmY2xvc2VHUSgmcm9icms7RidGaG8tRjA2LVEiOkYnRjNGNkY5RjtGPUY/RkFGQ0ZFRkgtRjA2LVEifkYnRjNGNkY5RjtGPUY/RkFGQy9GRlEmMC4wZW1GJy9GSUZocEZkcC1GLDYlUSJSRidGZm5GaW5GLy1GSzYkUSYyLjczNkYnRjMtRk82KEZbb0YzRmhvRltwRl5wRmhvRmFwThe potential at the origin is:print(); # input 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JSFH
JSFH
<Text-field style="Heading 1" layout="Heading 1">Question 3</Text-field>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(); # input placeholderTwo infinite insulating perpendicular planes are positioned on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeHpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeXpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn planes and charged by charge density per unit surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnNpZ21hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI6RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkk= What is the absulute value of the potential difference between the points 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 and 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print(); # input placeholder ?LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>Electrostatic problems are subject to the superpositon principle. we can solve the problem part after part and sum the results. We know the electric field outside an uniformly charged infinite plane, and thus we also know the potential. The electric field outside a single plane is 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 in the direction perpendicular to the plane. The potential is 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assuming the plane is on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeXpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn plane.Adding the potential of the second plane: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Subsituting the charge density: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print(); # input placeholderuse DocumentTools in
A := Typesetting[delayDotProduct](`<,>`(7, 7, 0), Unit('m'), true);
B := Typesetting[delayDotProduct](`<,>`(12, 12, 0), Unit('m'), true);
Phi := -sigma*x/(2*e_0)-sigma*y/(2*e_0);
Phi := subs({sigma = 7*Unit('picocoulomb'/'m'^2)}, Phi);
V := simplify(subs({x = (A(1)-B(1)), y = (A(1)-B(1)),e_0 = evalf(Constant(epsilon[0], units))}, Phi));
Do(%MathContainer0 = abs(V));
end use;
We were asked about the absulute value:
JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 4</Text-field>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JSFH
Two coaxial thin infinite cylindrical shells of radii 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 are charged by charge density per unit surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EtYCZzaWdtYTtfXzFgRictSSNtb0dGJDYtUSIsRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiNRLWAmc2lnbWE7X18yYEYnLUYwNi1RIjpGJ0YzRjYvRjpGOEY8Rj5GQEZCRkQvRkdRLDAuMjc3Nzc3OGVtRicvRkpGVA==respectivly.What is the potential diffrence between the two shells?
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>Once again, we know the electric field around an infinite cylindrical shell (you can find it using Gauss law). 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print(); # input placeholderthe resulting potential is 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print(); # input placeholderThe electric field of our setting is 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Since a particle traveling inside the gap between the shells will experience the field of the inner cylinder exclusively, we need to adress only the potential of the middle field region. The potential differnce is:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 5</Text-field>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A ball of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is charged by charge density 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print(); # input placeholderassuming that 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 is the potential at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJSRidGL0YyRjk= ?LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>We know that electric charge of spherical symmetry is seen from the outside as if it was concentrated at the center of the sphere. We will calculate the total charge within a ball of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJSRidGL0YyRjk= :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 = print(); # input placeholder using Gauss law: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkmbWZyYWNHRiQ2KC1GIzYjLUYsNiVRIlFGJ0YvRjItRiM2KC1JI21uR0YkNiRRIjRGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GW28tRiw2JVElJnBpO0YnL0YwRj1GOUZnbi1JJW1zdXBHRiQ2JS1GLDYlUSJyRidGL0YyLUYjNiMtRlo2JFEiMkYnRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNiNRMmAmdmFyZXBzaWxvbjtfXzBgRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmdwLyUpYmV2ZWxsZWRHRj0tRiw2I1EhRic= = print(); # input placeholderand the relation between the potential and the electric field:
JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 6</Text-field>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A slab of infinite length in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiekYnRi9GMkY5 directions and of thickness LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= direction is charged by a surface charge density of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSUYyat LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= and a charge density per unit volume QyQ+SSRyaG9HNiIsJiomSSJCR0YlIiIiSSJ4R0YlRilGKUkiQUdGJUYpISIi Assuming the potential is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. What is the potential at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJkRidGL0YyRjk= ?LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>Using Gauss law, the electric field inside the slab is composed of three parts. The electric field of the surface charge density, QyQ+SShFX19zdXJmRzYiKiZJJnNpZ21hR0YlIiIiLUkiL0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRJMCZ2YXJlcHNpbG9uO19fMEdGJSIiI0YoISIi the electric field of the left part of the slab (the part within LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRIjBGJ0Y+Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ ) QyQ+SShFX19MZWZ0RzYiKiYtSSIvRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEkwJnZhcmVwc2lsb247X18wR0YlIiIjIiIiLUkkaW50R0YlNiRJJHJob0dGJS9JInhHRiU7IiIhRjZGMCEiIg==print(); # input placeholder, and the electric field of the right part of the slab (the part between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJkRidGNEY3Rj5GPi8lJmNsb3NlR1EiXUYnRj4=) QyQ+SSlFX19SaWdodEc2IiomLUkiL0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRJMCZ2YXJlcHNpbG9uO19fMEdGJSIiIyIiIi1JJGludEdGKTYkSSRyaG9HRiUvSSJ4R0YlO0Y2SSJkR0YlRjAhIiI=print(); # input placeholderThe total electric field is the sum of the first two arguments, minus the third, as it's direction is negative.PkkiRUc2Ii1JKXNpbXBsaWZ5R0YkNiMsKEkoRV9fc3VyZkdGJCIiIkkoRV9fTGVmdEdGJEYqSSlFX19SaWdodEdGJCEiIg== = print(); # input placeholderThe electric potential is: