24. Electric Force & Field; Gauss' Law

Gauss' Law

Deriving Gauss's Law for Electric Flux via the Divergence Theorem from Vector Calculus

A hollow, conducting sphere with an outer radius of 0.250 m and an inner radius of 0.200 m has a uniform surface charge density of +6.37×10−6 C/m2. A charge of −0.500 μC is now introduced at the center of the cavity inside the sphere. (c) What is the electric flux through a spherical surface just inside the inner surface of the sphere?

Anderson Video - Gauss's Law Example

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area σ = 5.00×10−6 C/m2. (a) A small sphere of mass m = 8.00×10−6 kg and charge q is placed 3.00 cm above the sheet of charge and then released from rest. (a) If the sphere is to remain motionless when it is released, what must be the value of q?

The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately 7.4×10−15 m. (c) The electrons can be modeled as forming a uniform shell of negative charge. What net electric field do they produce at the location of the nucleus?

A conductor with an inner cavity, like that shown in Fig. 22.23c, carries a total charge of +5.00 nC. The charge within the cavity, insulated from the conductor, is −6.00 nC. How much charge is on (a) the inner surface of the conductor and (b) the outer surface of the conductor?

ΦC, ΦB, ΦA

ΦA, ΦB, ΦC

ΦA, ΦC, ΦB

ΦB, ΦC, ΦA

Rank the flux through surfaces A, B and C in the figure below from greatest to smallest.<img src="https://static.studychannel.pearsonprd.tech/courses/physics/thumbnails/images/48520df5"alt=""/>

Charge q is distributed uniformly throughout the volume of an insulating sphere of radius R = 4.00 cm. At a distance of r = 8.00 cm from the center of the sphere, the electric field due to the charge distribution has magnitude E = 940 N/C. What are (a) the volume charge density for the sphere?

The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately 7.4×10−15 m. (a) What is the electric field this nucleus produces just outside its surface?

Flux and Gauss Law

A very long insulating cylinder has radius <math display="inline"><mrow><mn>4.7</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math>. Every <math display="inline"><mrow><mn>3.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">m</mi></mrow></math> length of the cylinder contains a charge of <math display="inline"><mrow><mn>86</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">nC</mi></mrow></math> evenly distributed throughout. What is the strength of the electric field <math display="inline"><mrow><mn>3.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math> from the axis of the cylinder?

Gauss Law Cylinder, Infinite Line of Charge, Electric Flux & Field, Physics Problems

An infinitely long cylindrical conductor has radius r and uniform surface charge density σ. (b) In terms of σ, what is the magnitude of the electric field produced by the charged cylinder at a distance r > R from its axis? (c) Express the result of part (b) in terms of λ and show that the electric field outside the cylinder is the same as if all the charge were on the axis.

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area σ = 5.00×10−6 C/m2. (a) A small sphere of mass m = 8.00×10−6 kg and charge q is placed 3.00 cm above the sheet of charge and then released from rest. (b) What is q if the sphere is released 1.50 cm above the sheet?

You measure an electric field of 1.25×106 N/C at a distance of 0.150 m from a point charge. There is no other source of electric field in the region other than this point charge. (a) What is the electric flux through the surface of a sphere that has this charge at its center and that has radius 0.150 m? (b) What is the magnitude of this charge?

E4cm = 2.25 × 109 outwards; E12cm = 1.25 × 108 inwards

E4cm = 2.25 × 1013 outwards; E12cm = 1.25 × 1013 outwards

E4cm = 2.25 × 1013 outwards; E12cm = 6.25 × 1012 outwards

E4cm = 2.25 × 1013 outwards; E12cm = 1.25 × 1012 inwards

A spherical, conducting shell has a charge of –6C. If a 4C charge were placed at the center of the shell, what is the electric field at 4 cm? At 12 cm?<img src="https://static.studychannel.pearsonprd.tech/courses/physics/thumbnails/images/56639e5d"alt=""/>

A hollow, conducting sphere with an outer radius of 0.250 m and an inner radius of 0.200 m has a uniform surface charge density of +6.37×10−6 C/m2. A charge of −0.500 μC is now introduced at the center of the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere?

Charge q is distributed uniformly throughout the volume of an insulating sphere of radius R = 4.00 cm. At a distance of r = 8.00 cm from the center of the sphere, the electric field due to the charge distribution has magnitude E = 940 N/C. What are (b) the electric field at a distance of 2.00 cm from the sphere’s center?

An insulating sphere of radius <math display="inline"><mrow><mn>4.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math> is in the center of a conducting spherical shell with inner radius of <math display="inline"><mrow><mn>5.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math> and outer radius of <math display="inline"><mrow><mn>7.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math>. The insulating sphere has a charge of <math display="inline"><mrow><mo>&#x2212;</mo><mn>34</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">nC</mi></mrow></math> and the conducting spherical shell has a net charge of <math display="inline"><mrow><mn>42</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">nC</mi></mrow></math>. What is the total charge on the inner surface of the conducting spherical shell?

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of −3.63×1016 N·m2/C at the planet’s surface. Calculate: (b) the electric field at the planet’s surface (refer to the astronomical data inside the back cover);

Surface Charge Density

A very long conducting tube (hollow cylinder) has inner radius A and outer radius b. It carries charge per unit length +α, where α is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length +α. (b) What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?

A hollow, conducting sphere with an outer radius of 0.250 m and an inner radius of 0.200 m has a uniform surface charge density of +6.37×10−6 C/m2. A charge of −0.500 μC is now introduced at the center of the cavity inside the sphere. (b) Calculate the strength of the electric field just outside the sphere?

A sphere of radius <math display="inline"><mrow><mn>4.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math> has a charge of <math display="inline"><mrow><mn>23</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">nC</mi></mrow></math> evenly distributed throughout the sphere. What is the electric field strength <math display="inline"><mrow><mn>1.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math> from the center of the sphere?

Physics 37.1   Gauss's Law Understood (2 of 29) What is Electric Flux?

An infinitely long cylindrical conductor has radius r and uniform surface charge density σ. (a) In terms of σ and R, what is the charge per unit length λ for the cylinder?

An electric field with strength <math display="inline"><mrow><mn>3500</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">N</mi><mo>/</mo><mi mathvariant="normal" class="MathML-Unit">C</mi></mrow></math> exists just outside a face of a large block of aluminum. If the electric field points towards the block, what is the surface charge density on the face of the block?

There is not enough information to answer this question.

A cube has sides equal to <math display="inline"><mrow><mn>4.0</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">cm</mi></mrow></math>. The net flux through the cube is <math display="inline"><mrow><mn>12</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">N</mi><mo>&#x00B7;</mo><msup><mi mathvariant="normal" class="MathML-Unit">m</mi><mn>2</mn></msup><mo>/</mo><mi mathvariant="normal" class="MathML-Unit">C</mi></mrow></math> outward. Is the net charge in the cube positive, negative or zero?

Anderson Video - Gauss's Law Intro

A very long conducting tube (hollow cylinder) has inner radius A and outer radius b. It carries charge per unit length +α, where α is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length +α. (a) Calculate the electric field in terms of α and the distance r from the axis of the tube for (i) r < a; (ii) a < r < b; (iii) r > b. Show your results in a graph of E as a function of R.

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of −3.63×1016 N·m2/C at the planet’s surface. Calculate:(c) the charge density on Mars, assuming all the charge is uniformly distributed over the planet’s surface.

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 12.0 cm, giving it a charge of −49.0 μ C. Find the electric field (b) just outside the paint layer;

Using Gauss's Law to Calculate the Electric Flux and Charge in a Rectangular Prism

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of −3.63×1016 N·m2/C at the planet’s surface. Calculate: (a) the total electric charge on the planet;

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 12.0 cm, giving it a charge of −49.0 μ C. Find the electric field (a) just inside the paint layer;

Gauss's Law Problems - Conducting Sphere, Spherical Conductor, Electric Flux & Field, Physics

The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately 7.4×10−15 m. (b) What magnitude of electric field does it produce at the distance of the electrons, which is about 1.0×10−10 m?

Electric Field due to Hollow Shell

Electric Flux and Gauss’s Law | Electronics Basics #6

Electric Field Within Spherical Conductor

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