Gauss’s law II If a lightning bolt strikes a car (unless it is a convertible), electrons would quickly (in the order of nano seconds) move to the surface, so it is almost an ideal conductive shell that you can hide inside during a lightning storm!
Reading: Mazur 24.7-25.2 Deriving Gauss’s law Applying Gauss’s law Electric potential energy Electrostatic work
LECTURE 7 1 24.7 Deriving Gauss’s law Gauss’s law states that the flux through a closed surface is proportional to the enclosed charge, � , such that
� Φ = �
LECTURE 7 2 Question: 24.7-1 (Flux from point charge) Consider a spherical Gaussian surface of radius � with a point charge � at the center. � ⃗ The flux through a surface is given by Φ = ∮ � ⋅ ��. Using this definition for flux and Coulomb's law, calculate the flux through the Gaussian surface in terms of �, �, � and any constants.
LECTURE 7 3 Question: 24.7-1 (Flux from point charge) answer Consider a spherical Gaussian surface of radius � with a point charge � at the center. � ⃗ The flux through a surface is given by Φ = ∮ � ⋅ ��. Using this definition for flux and Coulomb's law, calculate the flux through the Gaussian surface in terms of �, �, � and any constants. The electric field is always normal to the Gaussian surface and has equal magnitude at all points on the surface. Therefore � Φ = � ⋅ ��⃗ = ��� = � �� = � 4�� = � 4�� = 4��� � From Gauss’s law Φ = Therefore Φ = = 4���, and � =
LECTURE 7 4 Reading quiz (Problem 24.70) There is a Gauss's law for gravity analogous to Gauss's law for electricity. If the electric flux through a closed surface is proportional to the enclosed charge, what is the gravitational flux proportional to? A. acceleration B. energy C. force D. mass Write an equation for Gauss's law for gravity for the mass �. Express your answer in terms of the variable �, the gravitational constant �, and �.
Φ = −4��� Φ = ∮ �⃗ ⋅ ��⃗ = ∮ −��� = −� ∮ �� = −� 4�� = − � 4�� = −4���
LECTURE 7 5 Question: 24.7-2 (Flux from uniformly charged rod)
An infinitely long charged rod has uniform charge density of λ, and passes through a cylinder (gray). The cylinder in case 2 has twice the radius and half the length compared to the cylinder in case 1. What is the ratio of flux through the cylinder in case 1 over the flux through the cylinder in case 2, , ? ,
LECTURE 7 6 Question: 24.7-2 (Flux from uniformly charged rod) answer
An infinitely long charged rod has uniform charge density of λ, and passes through a cylinder (gray). The cylinder in case 2 has twice the radius and half the length compared to the cylinder in case 1. What is the ratio of flux through the cylinder in case 1 over the flux through the cylinder in case 2, , ? , 2