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![Page 1: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/1.jpg)
A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell is -3 Q, and it is insulated from its surroundings. Derive expressions for the electric field magnitude E in terms of the distance from the center r for the regions r<a, a<r<b, and r>b.
a b
-3Q+Q
Bold type denotes vector quantities
Bold type denotes vector quantities
![Page 2: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/2.jpg)
• Question 1• Question 2• Question 3• Question 4• Question 5• Question 6• Question 7• Question 8• Question 9
![Page 3: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/3.jpg)
• A : I• B : II & IV• C : III• D : III & IV
1. Which of the following physics principles should one use to solve this problem?
I. Ampere’s LawII. Faraday’s Law of InductionIII. Gauss’s LawIV.Superposition of Electric Fields
![Page 4: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/4.jpg)
This law deals with magnetic fields produced by electric current.
Choice: A
Incorrect
![Page 5: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/5.jpg)
Faraday’s law deals with the time rate of change of magnetic flux, so this is not applicable to our situation.
On the other hand, the principle of superposition of electric fields is very helpful here. Considering the
contributions to the electric field from each charge will make this task easier to evaluate.
Choice: B
Incorrect
![Page 6: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/6.jpg)
We can use this law to solve for E as we exploit symmetry, but another
physics principle will also be helpful.
Choice: C
Incorrect
![Page 7: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/7.jpg)
Choice: DCorrect
Gauss’s law and the principle of superposition of electric fields are very helpful here. Considering the
contributions to the electric field from the charge in each region will make this task easier to evaluate.
![Page 8: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/8.jpg)
2. Which statement correctly describes Gauss’s Law?
• A : The total electric field through a closed surface is equal to the net charge inside the surface.
• B : The total electric flux through a closed surface is equal to the total charge inside the surface divided by the area.
• C : The total electric flux through a closed surface is equal to the total charge inside the surface divided by o (permittivity of free space).
![Page 9: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/9.jpg)
Gauss’s Law relates the electric flux through a surface to the total charge enclosed by the surface.
Choice: A
Incorrect
€
E ≠Qenc
![Page 10: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/10.jpg)
Choice: B
Incorrect
According to Gauss’s Law, the total electric flux through a closed surface is
equal to the total charge inside the surface divided by
o , not by the area.
![Page 11: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/11.jpg)
Choice: C
Correct
€
Φnet = E • dA∫ =Qenc
o
In mathematical form, Gauss’s Law is expressed as:
![Page 12: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/12.jpg)
3. What is a convenient Gaussian surface for this system?
• A: circle• B: cube• C: sphere
![Page 13: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/13.jpg)
Choice: A
Incorrect
The system is three dimensional.
![Page 14: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/14.jpg)
The magnitude of the electric field is not the same at all points
on a cubical surface.
Choice: B
Incorrect
![Page 15: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/15.jpg)
Choice: C
Correct
This is a good choice, because the system shows spherical
symmetry.
![Page 16: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/16.jpg)
Please get out a piece of scratch paper and sketch the charge configuration of the situation described in the problem statement. For now, only draw the arrangement due to the charge +Q inside the cavity. Draw your Gaussian surface with a radius r<a.
![Page 17: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/17.jpg)
Example: Your sketch should look something like this:
a
r+Q-
-
--
-
-
- --
-Q+
+
+
+
+
+
++
+
+Q
1. A negative charge of magnitude -Q is induced uniformly around the inner surface of the cavity.
1. A negative charge of magnitude -Q is induced uniformly around the inner surface of the cavity.
2. The negative charge is drawn to the inner surface and a positive charge remains on the outer surface of the conducting shell.
2. The negative charge is drawn to the inner surface and a positive charge remains on the outer surface of the conducting shell.
Gaussian surface
Gaussian surface
![Page 18: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/18.jpg)
4. For r<a, as depicted in our sketch, the field strength is equal in magnitude everywhere on the surface and is radially outward in direction. Choose all of the following that are correct expressions for the total flux in this case.
I)
II)
III)
A: I only
C: III only
B: II only
D: I and II only
€
E4πr2
€
+Qo
€
+Q4πr2o
E: I and III only
![Page 19: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/19.jpg)
This is a correct expression, but it is not the only one.
Choice: A
Incorrect
![Page 20: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/20.jpg)
This is a correct expression, but it is not the only one.
Choice: B
Incorrect
![Page 21: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/21.jpg)
Check the units. This is an expression for electric field.
Choice: C
Incorrect
![Page 22: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/22.jpg)
Both these expressions are correct.
Choice: D
Correct
€
Φnet =+Qo
€
Φnet = E • dA∫ =E dA∫ =EA =E4πr2
Since the electric field is perpendicular to the infinitesimal area dA, the magnitude of the electric field E can be pulled out of the integral.
Mathematical expression of Gauss’s Law:
The total charge enclosed in our Gaussian surface is only +Q, therefore:
€
Φnet = E • dA∫ =Qenc
o
![Page 23: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/23.jpg)
One of these expressions is not correct. Try Again.
Choice: E
Incorrect
![Page 24: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/24.jpg)
5. Which one of the following expressions describes the electric field E in the region
r<a?
• A:
• B:
• C:
€
−Q4πr2o
€
+Q4πr2o
€
−3Q4πr2o
![Page 25: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/25.jpg)
Choice: A
Incorrect
The total charge enclosed in the Gaussian sphere is +Q.
![Page 26: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/26.jpg)
Choice: B
Correct
€
E4πr2 =+Qo
E = +Q4πr2o
From the previous question we have:
Simple algebra shows that:
![Page 27: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/27.jpg)
Choice: C
Incorrect
Since our Gaussian surface has a radius r less than a, the only charge that is enclosed is the
point charge +Q.
![Page 28: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/28.jpg)
6. For electrostatic equilibrium, the electric field inside the conductor (metal) is which of the following?
• A: uniform but non-zero• B: zero• C: non-uniform
![Page 29: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/29.jpg)
Choice: A
Incorrect
This is in violation of the equilibrium condition.
![Page 30: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/30.jpg)
Choice: B
Correct
This implies equilibrium.
![Page 31: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/31.jpg)
Choice: C
Incorrect
The charges are uniformly distributed.
![Page 32: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/32.jpg)
• Please make another sketch of the charge configuration of the spherical shell, this time only consider the -3Q charge that is placed on the outer surface of the conductor.
![Page 33: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/33.jpg)
Example
-
-
-
-
-
-
-
--
-
-3Q
The charge -3Q distributes uniformly on the surface of the conductor, making the field everywhere inside of the conductor equal to zero.
The charge -3Q distributes uniformly on the surface of the conductor, making the field everywhere inside of the conductor equal to zero.Notice that
there is no charge enclosed if the gaussian surface is placed anywhere inside of the surface of the conductor.
Notice that there is no charge enclosed if the gaussian surface is placed anywhere inside of the surface of the conductor.
![Page 34: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/34.jpg)
• Considering a superposition of the electric fields produced by the point charge +Q and the surface charge -3Q will help us find expressions for the electric field in the regions a<r<b and r>b.
• Sketch the charge configuration of the spherical shell considering all sources of electric field (combine your two previous drawings).
![Page 35: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/35.jpg)
Superposition of electric fields and charge summation
+ =-
--
-
-
--
+
+
+
+
+
+
+Q
++Q
-Q
+Q
-Q
-3Q
+Q
-
--
--
--
-
-
- -
-
-
-
-
- -
+
+
++
The red arrows depict electric field lines. Notice that there are more field lines in the second drawing pointing inwards than there are in the first pointing outwards.
The red arrows depict electric field lines. Notice that there are more field lines in the second drawing pointing inwards than there are in the first pointing outwards.
![Page 36: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/36.jpg)
7. Add a Gaussian surface to your drawing with a radius a<r<b.
What is the electric field in this region (inside of the conductor).
• A: E=0
• B: E=
• C: E=
€
+Q4πr2o
€
−Q4πr2o
![Page 37: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/37.jpg)
Choice: A
Correct
€
E4πr2 =Qenc
o
Qenc =(+Q) + (−Q) = 0
E = 0
The electric field inside of a conductor is always equal to zero.
r
a
b
![Page 38: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/38.jpg)
Choice: B
Incorrect
Remember that the electric field inside of a conductor is always zero.
Your Gaussian surface should enclose more than just the point
charge +Q.
![Page 39: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/39.jpg)
Remember that the electric field inside of a conductor is always zero.
Your Gaussian surface should enclose more than just the induced charge -Q on the inside of the shell.
Choice: C
Correct
![Page 40: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/40.jpg)
8. What is the net charge enclosed by a Gaussian sphere in the r>b region?
• A: +Q
• B: -3Q
• C: -Q
• D: -2Q
![Page 41: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/41.jpg)
Choice: A
Incorrect
br
€
Qenc =(+Q) + (−Q) + (+Q) + (−3Q)Qenc = −2Q
Gaussian surface
Gaussian surface
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
![Page 42: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/42.jpg)
Choice: B
Incorrect
br
€
Qenc =(+Q) + (−Q) + (+Q) + (−3Q)Qenc = −2Q
Gaussian surface
Gaussian surface
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
![Page 43: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/43.jpg)
Choice: C
Incorrect
br
€
Qenc =(+Q) + (−Q) + (+Q) + (−3Q)Qenc = −2Q
Gaussian surface
Gaussian surface
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
![Page 44: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/44.jpg)
Choice: D
Correct
br
€
Qenc =(+Q) + (−Q) + (+Q) + (−3Q)Qenc = −2Q
Gaussian surface
Gaussian surface
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
The point charge in the center, the induced charges -Q and +Q on the inner and outer surfaces of the shell, and the -3Q spread around the outside of the shell are all enclosed by the Gaussian sphere.
![Page 45: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/45.jpg)
9. Which one of the following is the expression for the electric field E in the region r>b?
A: (radially outward)
C: (radially inward)
B: (radially outward)
D: (radially inward)
€
+Q(4πr2εo )
€
−Q2πr2o
€
−2Qo
€
−Q2πr2o
![Page 46: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/46.jpg)
The total enclosed charge is not +Q. Also, field lines from a negative
source charge are directed inward.Please check your sketch and try
again.
Choice: A
Incorrect
![Page 47: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/47.jpg)
Choice: B
Incorrect
Field lines from a negative source charge are directed inward.
![Page 48: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/48.jpg)
Field lines from a negative source charge are directed inward.
Choice: C
Correct
€
E =Qenc
4πr2ε0=−2Q
4πr2ε0=−Q
2πr2ε0
Reasoning:
![Page 49: A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell.](https://reader033.fdocuments.in/reader033/viewer/2022061305/55143173550346d8488b5fd2/html5/thumbnails/49.jpg)
Choice: D
Incorrect
This is the total flux through the Gaussian sphere.