Consider two conducting spheres with radii R 1 and R 2 separated by a distance much greater than cither radius. A total charge Q is shared between the spheres. We wish to show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero. The total charge Q is equal to q 1 + q 2 , where q 1 represents the charge on the first sphere and q 2 the charge on the second. Because the spheres are very far apart, you can assume the charge of each is uniformly distributed over its surface. (a) Show that the energy associated with a single conducting sphere of radius R and charge q surrounded by a vacuum is U E = k e q 2 /2 R . (b) Find the total energy’ of the system of two spheres in terms of the total charge Q , and the radii and R 1 and R 2 . (c) To minimize the energy, differentiate the result to part (b) with respect to q 1 and set the derivative equal to zero. Solve for q 1 in terms of Q and the radii. (d) From the result to part (c), find the charge q 2 . (e) Find the potential of each sphere. (f) What is the potential difference between the spheres?
Consider two conducting spheres with radii R 1 and R 2 separated by a distance much greater than cither radius. A total charge Q is shared between the spheres. We wish to show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero. The total charge Q is equal to q 1 + q 2 , where q 1 represents the charge on the first sphere and q 2 the charge on the second. Because the spheres are very far apart, you can assume the charge of each is uniformly distributed over its surface. (a) Show that the energy associated with a single conducting sphere of radius R and charge q surrounded by a vacuum is U E = k e q 2 /2 R . (b) Find the total energy’ of the system of two spheres in terms of the total charge Q , and the radii and R 1 and R 2 . (c) To minimize the energy, differentiate the result to part (b) with respect to q 1 and set the derivative equal to zero. Solve for q 1 in terms of Q and the radii. (d) From the result to part (c), find the charge q 2 . (e) Find the potential of each sphere. (f) What is the potential difference between the spheres?
Consider two conducting spheres with radii R1 and R2 separated by a distance much greater than cither radius. A total charge Q is shared between the spheres. We wish to show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero. The total charge Q is equal to q1 + q2, where q1 represents the charge on the first sphere and q2 the charge on the second. Because the spheres are very far apart, you can assume the charge of each is uniformly distributed over its surface. (a) Show that the energy associated with a single conducting sphere of radius R and charge q surrounded by a vacuum is UE = keq2/2R. (b) Find the total energy’ of the system of two spheres in terms of the total charge Q, and the radii and R1 and R2. (c) To minimize the energy, differentiate the result to part (b) with respect to q1 and set the derivative equal to zero. Solve for q1 in terms of Q and the radii. (d) From the result to part (c), find the charge q2. (e) Find the potential of each sphere. (f) What is the potential difference between the spheres?
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(a)
Expert Solution
To determine
To show: The energy associated with a single conducting sphere is UE=keq22R.
Answer to Problem 26.40P
The energy associated with a single conducting sphere is UE=keq22R.
Explanation of Solution
Given info: The radii of two conducting sphere is R1 and R2. The total charge shared between them is Q. The charge on first and second sphere is q1 and q2 respectively.
Write the expression to calculate the capacitance of a sphere of radius R.
C=Rke
Here,
ke is the Coulomb’s law constant.
Write the expression to calculate the potential difference.
ΔV=keqR
Here,
ΔV is the potential difference of the capacitor.
q is the charge on a single sphere.
Write the expression to calculate the energy stored in the capacitor.
UE=12C(ΔV)2
Substitute Rke for C and keqR for ΔV in above equation.
UE=12(Rke)(keqR)2=keq22R
Conclusion:
Therefore, the energy associated with a single conducting sphere is UE=keq22R.
(b)
Expert Solution
To determine
The total energy of the system of two spheres in term of q1, Q and the radii R1 and R2.
Answer to Problem 26.40P
The total energy of the system of two spheres in term of q1, Q and the radii R1 and R2 is 12keq12R1+ke2(Q−q1)2R2.
Explanation of Solution
Given info: The radii of two conducting sphere is R1 and R2. The total charge shared between them is Q. The charge on first and second sphere is q1 and q2 respectively.
Write the expression to calculate the capacitance of a sphere of radius R.
C=Rke
Write the expression to calculate the total energy of the system of two sphere.
UE=12q12C1+12q22C2
Substitute R1ke for C1 and R2ke for C2 in above equation.
UE=12q12(R1ke)+12q22(R2ke) (1)
The sum of charge of both sphere are,
Q=q1+q2q2=Q−q1
Substitute Q−q1 for q2 in above equation.
UE=12keq12R1+ke2(Q−q1)2R2
Thus, the total energy of the system of two spheres in term of q1, Q and the radii R1 and R2 is 12keq12R1+ke2(Q−q1)2R2.
Conclusion:
Therefore, the total energy of the system of two spheres in term of q1, Q and the radii R1 and R2 is 12keq12R1+ke2(Q−q1)2R2.
(c)
Expert Solution
To determine
The value of q1 by differentiating the result of part (b).
Answer to Problem 26.40P
The value of q1 is q1=R1QR1+R2.
Explanation of Solution
Given info: The radii of two conducting sphere is R1 and R2. The total charge shared between them is Q. The charge on first and second sphere is q1 and q2 respectively.
The total energy of the system of two spheres in term of q1, Q and the radii R1 and R2 is,
UE=12keq12R1+ke2(Q−q1)2R2.
Differentiate the above equation with respect to q1 and equate to zero.
Given info: The radii of two conducting sphere is R1 and R2. The total charge shared between them is Q. The charge on first and second sphere is q1 and q2 respectively.
The value of q1 is,
q1=R1QR1+R2.
The sum of charge of both sphere are,
Q=q1+q2q2=Q−q1
Substitute R1QR1+R2 for q2 in above equation.
q2=Q−R1QR1+R2=R2QR1+R2
Conclusion:
Therefore, the value of q2 is q2=R2QR1+R2.
(e)
Expert Solution
To determine
The potential of each sphere.
Answer to Problem 26.40P
The potential of each sphere is keQR1+R2.
Explanation of Solution
Given info: The radii of two conducting sphere is R1 and R2. The total charge shared between them is Q. The charge on first and second sphere is q1 and q2 respectively.
Write the expression to calculate the potential of first sphere.
V1=keq1R1
Substitute R1QR1+R2 for q1 in above equation.
V1=keR1×R1QR1+R2=keQR1+R2
Write the expression to calculate the potential of second sphere.
V2=keq2R2
Substitute R2QR1+R2 for q1 in above equation.
V2=keR2×R2QR1+R2=keQR1+R2
Thus, the potential of each sphere is keQR1+R2.
Conclusion:
Therefore, the potential of each sphere is keQR1+R2.
(f)
Expert Solution
To determine
The potential difference between the spheres.
Answer to Problem 26.40P
The potential difference between the spheres is zero.
Explanation of Solution
Given info: The radii of two conducting sphere is R1 and R2. The total charge shared between them is Q. The charge on first and second sphere is q1 and q2 respectively.
The potential difference is,
ΔV=V1−V2
Substitute keQR1+R2V1andV2 in above equation.
ΔV=keQR1+R2−keQR1+R2=0
Conclusion:
Therefore, the potential difference between the spheres is zero.
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