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What is the net force on a test charge Q at the center? The force on Q must be 0 by symmetry. The x, y, and z components of the fields due to each charge cancel, so the force vanishes.
What is the force on Q? Explain your reasoning carefully. Here, r is the distance from the center to each charge.
What is the force on a test charge Q at the center. Again, by symmetry, the answer is still 0. The same answer as in part b, equation 1, and for the same reason.
The method is the same as in problem 2. This part is simple. Do it in two different ways. What is the flux of E through one side?
We need to surround the charge symmetrically, since otherwise it is unclear what fraction of the flux is going through each face. For instance, no flux goes through the three faces adjacent to the charge, since they are parallel to the electric field.
However, if make a larger cube, one centered on the charge, we can determine the flux through each face by symmetry. If we center the cube on the charge, then we are effectively adding 7 cubes of the same dimensions as the original cube, yielding 8 total cubes.
Use infinity as your reference point.
Compute the gradient of V in each eletromanyetik, and check that it yields the correct result. Compute the gradient of your potential, and check that it yields the correct field. Instead we integrate from s to some point a.
It takes no work to bring in the first charge. Therefore, tfori work for the total configuration is: The shell carries no net charge. How do your answers to a and b change? If the outer surface is grounded, the charge on the outer surface will go to ground, since it will go from a region of high potential to region of low potential.
As a result, the potential at the center will have no contribution from the outer part of the dlektromanyetik. At the center of each cavity a point charge is placed — call these charges qa and qb.
The charge at the middle of each cavity will induce an equal and opposite charge density on the cavity epektromanyetik. Let the radii of the cavities be a and b, respectively. Therefore, outside of R: The electric field is due to the charges themselves, and the field due to charge a does not influence charge b, and vice versa.
Eletrodinmica – David J. Griffiths – 3 Edio
Therefore the force on each charge is 0. Furthermore, it would affect ERsince the new charge contributes its own electric field. This potential can be decomposed into a linear combination of Legendre polynomials: So now I have the equality if I replace P3 and P1 with the explicit forms of the polynomials: Find the potential inside and outside the cylinder. If we note that all the coefficients vanish except b5 and d5: Find the approximate potential for points on the z axis, far from the sphere.
Find a simple approximate formula for the potential valid at points far from the origin. Express your answer in spherical coordinates. To first order, the monopole term the potential vanishes, since the total charge is zero.
For each of the arrangements below, find i the monopole moment, ii the dipole moment, and iii the approximate potential in spherical coordinates at large r include both the monopole and dipole contributions. The monopole moment is that total charge, which is again 2q. Find the approximate electric field at points far from the origin. Express your answer in spherical coordinates, and include the two lowest orders in the multipole expansion.
We need to find the electric field everywhere first. First pretend the field inside is just E0and use Eq. This polarization generates a field of its own, E1 Ex. Sum the series, and compare your answer with Eq. The field resulting from this polarization is given by equation 4. Find the energy of this configuration. We will use equation 4. Find all the bound charges, and check that they add up to zero. Now we need to find the total charge: What is the total bound charge on the surface?
Where is the compensating negative bound charge located? Find the force on a square loop side alying in the yz plane and centered at the origin, if it carries a current I, flowing counterclockwise, when you look down the x axis. The force on the two horizontal segments in z field flips sign.
How great would v have to be in order for the magnetic attraction to balance the electric repulsion? The inner solenoid radius a has n1 turns per unit length, and the outer one radius b has n2. Find B in each of the three regions: Outside, the field is zero. Thus, in region iwe have contributions from both the larger and the smaller solenoid. In region iithe only contribution to the magnetic field is from the larger solenoid.
The region is outside the smaller solenoid, so the magnetic field due to the smaller one is zero.
According to example 5. How is this derived? Thus, above and below the capacitor, the field is zero, since the surface current K is pointed in elektromanyehik direction due to the opposite charge on the plates.
In between the capacitor plates, the field adds: Equating this force with the force per unit area in part b yields: Thus, the forces would never balance. If the square loop is free to rotate, what will its equilibrium orientation be? Grifffiths the magnetic field due to M inside and outside the cylinder. Find the magnetic field due to M, for points inside and outside the cylinder.
The easiest way to do this is to calculate the bound currents: Since the surface current is Find the magnetic field inside and outside the cylinder by two different methods: Notice that the second method is much faster, and avoids any explicit reference to the bound currents. A current I flows down the inner conductor and returns along the outer one; in each case the current distributes itself uniformly over the surface Fig.
Ele,tromanyetik the magnetic field in the region between the tubes. As a check, calculate the magnetization and the bound currents, and confirm that together, of course, with the free currents they generate the correct field.
Find all the bound currents. What is the net bound Rcurrent flowing down the wire. A resistor R is connected across the rails and a uniform magnetic field B, pointing into the page, fills the elekhromanyetik region. In what direction does it flow? If the bar is moving to the right, that means that the flux through the loop is increasing.
Since the current is up through the bar and the magnetic field is into the page, the force is to the left, as it must be.