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located at (d/2,0,0), another + located at 2. A charge distribution has 3 point charges: +...
Q1. MULTIPOLES - point charges You haye four point charges. Their location and charges in Cartesian...
Q1. MULTIPOLES - point charges You haye four point charges. Their location and charges in Cartesian coordinates are: A positive charge -2q located at (a,0,0), another charge -2q located at (-2,0,0), a 3rd charge -q located at (0,0,b), and finally a fourth charge +57 located at (0,0,-b) - What is the total charge, and dipole moment, of this distribution of charges? Use the methods of "the multipole expansion" (Griffiths section 3.4.1) to find a simple approximate formula for V(r,0) (in...
4. A spherically sym metric charge distribution has the following radial dependence for the volume charge...
4. A spherically sym metric charge distribution has the following radial dependence for the volume charge density ρ 0 if r > R where γ is a constant a) What units must the constant y have? b) Find the total charge contained in the sphere of radius R centered at the origin. c) Use the integral form of Gauss's law to determine the electric field in the region r < R. (Hint: if the charge distribution is spherically symmetric, what...
4 A spherically symmetric charge distribution has the following radial dependence for the volume charge density...
4 A spherically symmetric charge distribution has the following radial dependence for the volume charge density ρ: 0 if r R where γ is a constant a) What units must the constant γ have? b) Find the total charge contained in the sphere of radius R centered at the origin c) Use the integral form of Gauss's law to determine the electric field in the region r R. (Hint: if the charge distribution is spherically symmetric, what can you say...
Problem 2 - Point charge and plane (20 pts) A point charge q (q>0) is located...
Problem 2 - Point charge and plane (20 pts) A point charge q (q>0) is located a distance d above an infinite conducting plane lying in the x-y plane. The plane is connected to the ground (Fig.1), so that the electric potential V at any point on the plane satisfies V=0. Calculating the electric potential generated by the point charge-grounded plane combination at any point P is more complicated than it looks because the conducting plane pulls some electric charge...
Only part f) please! 4 A spherically symmetric charge distribution has the following radial dependence for...
Only part f) please!
4 A spherically symmetric charge distribution has the following radial dependence for the volume charge density ρ ρ(r) If r > R where y is a constant a) What units must the constant y have? b) Find the total charge contained in the sphere of radius R centered at the origin c) Use the integral form of Gauss's law to determine the electric field in the region r < R. Hint: if the charge distribution is...
d) Three point charges are located as shown in Figure 2 (see page 6). All three...
d) Three point charges are located as shown in Figure 2 (see
page 6). All three charges are
a distance a from the origin.
i) Calculate the dipole moment for this charge configuration
[2]
ii) By calculating the first two terms of the multipole expansion
for this cofiguration,f ind
an approximate expression for the scalar potential of this
cofiguration. Express your
result in spherical polar coordinates. [5]
iii) Hence provide an approximate expression for the electrical
field, E, for this...
You have four point charges. Their location and charges in Cartesian coordinates are: a positive charge,...
You have four point charges. Their location and charges in Cartesian coordinates are: a positive charge, 2q, located at (a,0,0), another charge -2q located at (-a,0,0), a 3rd charge -q located at (0,0,b), and finally a fourth charge +5q located at (0,0,-b) (a) What is the total charge, and dipole moment, of this distribution of charges? Use the methods of "the multipole expansion" (Griffiths section 3.4.1) to find a simple approximate formula for V(r,0) (in spherical coordinates!) valid at points...
2. (8 pts) Consider the screened Coulomb potential of a point charge q that arises e.g....
2. (8 pts) Consider the screened Coulomb potential of a point charge q that arises e.g. in plasma physics: V(r) = kq xPL T/A), where is a constant called screening length. a) (2 pts) Determine E(r) associated with potential. b) (2 pts) Find the charge distribution p(r) that produces this potential. (Think carefully about what happens at the origin!) c) (2 pts) Show by explicit integration over p(r) that the charge represented by this distribution is zero. (If you don't...
Suppose an individual’s utility function for two goods X and Y is givenby U(X,Y) = X^(3/4)Y^(1/4)...
Suppose an individual’s utility function for two goods X and Y is givenby U(X,Y) = X^(3/4)Y^(1/4) Denote the price of good X by Px, price of good Y by Py and the income of the consumer by I. a) (2 points) Write down the budget constraint for the individual. b) (4 points) Derive the marginal utilities of X and Y. c) (3 points) Derive the expression for the marginal rate of substitution of X for Y. Write down the tangency...
3. (20) A spherically symmetric charge distribution creates the following electric field (2) E E,r with...
3. (20) A spherically symmetric charge distribution creates the following electric field (2) E E,r with 20 r r < a for 4meoa3 (3) E,= Q 4mor2 for r> a where Q and a are positive constants of suitable units. (a) Draw a graph of E, for 0 <r3a; please label your graph clearly (b) Calculate the charge distribution that generates this electric field. (c) Draw a graph of the charge distribution for 0 <r< 3a; please label your graph...
Q1. MULTIPOLES - point charges You haye four point charges. Their location and charges in Cartesian coordinates are: A positive charge -2q located at (a,0,0), another charge -2q located at (-2,0,0), a 3rd charge -q located at (0,0,b), and finally a fourth charge +57 located at (0,0,-b) - What is the total charge, and dipole moment, of this distribution of charges? Use the methods of "the multipole expansion" (Griffiths section 3.4.1) to find a simple approximate formula for V(r,0) (in...
4. A spherically sym metric charge distribution has the following radial dependence for the volume charge density ρ 0 if r > R where γ is a constant a) What units must the constant y have? b) Find the total charge contained in the sphere of radius R centered at the origin. c) Use the integral form of Gauss's law to determine the electric field in the region r < R. (Hint: if the charge distribution is spherically symmetric, what...
4 A spherically symmetric charge distribution has the following radial dependence for the volume charge density ρ: 0 if r R where γ is a constant a) What units must the constant γ have? b) Find the total charge contained in the sphere of radius R centered at the origin c) Use the integral form of Gauss's law to determine the electric field in the region r R. (Hint: if the charge distribution is spherically symmetric, what can you say...
Problem 2 - Point charge and plane (20 pts) A point charge q (q>0) is located a distance d above an infinite conducting plane lying in the x-y plane. The plane is connected to the ground (Fig.1), so that the electric potential V at any point on the plane satisfies V=0. Calculating the electric potential generated by the point charge-grounded plane combination at any point P is more complicated than it looks because the conducting plane pulls some electric charge...
Only part f) please!
4 A spherically symmetric charge distribution has the following radial dependence for the volume charge density ρ ρ(r) If r > R where y is a constant a) What units must the constant y have? b) Find the total charge contained in the sphere of radius R centered at the origin c) Use the integral form of Gauss's law to determine the electric field in the region r < R. Hint: if the charge distribution is...
d) Three point charges are located as shown in Figure 2 (see
page 6). All three charges are
a distance a from the origin.
i) Calculate the dipole moment for this charge configuration
[2]
ii) By calculating the first two terms of the multipole expansion
for this cofiguration,f ind
an approximate expression for the scalar potential of this
cofiguration. Express your
result in spherical polar coordinates. [5]
iii) Hence provide an approximate expression for the electrical
field, E, for this...
You have four point charges. Their location and charges in Cartesian coordinates are: a positive charge, 2q, located at (a,0,0), another charge -2q located at (-a,0,0), a 3rd charge -q located at (0,0,b), and finally a fourth charge +5q located at (0,0,-b) (a) What is the total charge, and dipole moment, of this distribution of charges? Use the methods of "the multipole expansion" (Griffiths section 3.4.1) to find a simple approximate formula for V(r,0) (in spherical coordinates!) valid at points...
2. (8 pts) Consider the screened Coulomb potential of a point charge q that arises e.g. in plasma physics: V(r) = kq xPL T/A), where is a constant called screening length. a) (2 pts) Determine E(r) associated with potential. b) (2 pts) Find the charge distribution p(r) that produces this potential. (Think carefully about what happens at the origin!) c) (2 pts) Show by explicit integration over p(r) that the charge represented by this distribution is zero. (If you don't...
3. (20) A spherically symmetric charge distribution creates the following electric field (2) E E,r with 20 r r < a for 4meoa3 (3) E,= Q 4mor2 for r> a where Q and a are positive constants of suitable units. (a) Draw a graph of E, for 0 <r3a; please label your graph clearly (b) Calculate the charge distribution that generates this electric field. (c) Draw a graph of the charge distribution for 0 <r< 3a; please label your graph...
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