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free/or 0.5$ |
| 661 | A sphere of radius r carries a surface charge of density σ = ar, where a is a constant vector, and r is the radius vector of a point of the sphere relative to its centre. Find the electric field strength vector at the centre of the sphere. |
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| 662 | Suppose the surface charge density over a sphere of radius R depends on a polar angle θ as σ = σ0 cos θ, where σ0 is a positive constant. Show that such a charge distribution can be represented as a result of a small relative shift of two uniformly charged balls of radius R whose charges are equal in magnitude and opposite in sign. Resorting to this representation, find the electric field strength vector inside the given sphere. |
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| 663 | Find the electric field strength vector at the centre of a ball of radius R with volume charge density ρ = ar, where a is a constant vector, and r is a radius vector drawn from the ball's centre. |
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| 664 | A very long uniformly charged thread oriented along the axis of a circle of radius R rests on its centre with one of the ends. The charge of the thread per unit length is equal to λ. Find the flux of the vector E across the circle area. |
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| 665 | Two point charges q and -q are separated by the distance 2l (Fig. 3.3). Find the flux of the electric field strength vector across a circle of radius R. |
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| 666 | A ball of radius R is uniformly charged with the volume density ρ. Find the flux of the electric field strength vector across the ball's section formed by the plane located at a distance r0 < R from the centre of the ball. |
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| 667 | Each of the two long parallel threads carries a uniform charge λ per unit length. The threads are separated by a distance l. Find the maximum magnitude of the electric field strength in the symmetry plane of this system located between the threads. |
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| 668 | An infinitely long cylindrical surface of circular crosssection is uniformly charged lengthwise with the surface density s = s0cosf, where p i s the polar angle of the cylindrical coordinate system whose z axis coincides with the axis of the given surface. Find the magnitude and direction of the electric field strength vector on the z axis. |
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| 669 | The electric field strength depends only on the x and y coordinates according to the law E = a(xi + yj)/(x2 + y2), where a is a constant, i and j are the unit vectors of the x and y axes. Find the flux of the vector E through a sphere of radius R with its centre at the origin of coordinates. |
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| 670 | A ball of radius R carries a positive charge whose volume density depends only on a separation r from the ball's centre as ρ = ρ0(1 - r/R), where ρ0 is a constant. Assuming the permittivities of the ball and the environment to be equal to unity, find: (a) the magnitude of the electric field strength as a function of the distance r both inside and outside the ball; (b) the maximum intensity Emax and the corresponding distance rm. |
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| 671 | A system consists of a ball of radius R carrying a spherically symmetric charge and the surrounding space filled with a charge of volume density ρ = α/r, where α is a constant, r is the distance from the centre of the ball. Find the ball's charge at which the magnitude of the electric field strength vector is independent of r outside the ball. How high is this strength? The permittivities of the ball and the surrounding space are assumed to be equal to unity. |
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| 672 | A space is filled up with a charge with volume density ρ = ρ0e-αr3, where ρ0 and α are positive constants, r is the distance from the centre of this system. Find the magnitude of the electric field strength vector as a function of r. Investigate the obtained expression for the small and large values of r, i.e. at αr3 << 1 and αr3 >> 1. |
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| 673 | Inside a ball charged uniformly with volume density ρ there is a spherical cavity. The centre of the cavity is displaced with respect to the centre of the ball by a distance a. Find the field strength E inside the cavity, assuming the permittivity equal to unity. |
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| 674 | Inside an infinitely long circular cylinder charged uniformly with volume density p there is a circular cylindrical cavity. The distance between the axes of the cylinder and the cavity is equal to a. Find the electric field strength E inside the cavity. The permittivity is assumed to be equal to unity. |
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| 675 | There are two thin wire rings, each of radius R, whose axes coincide. The charges of the rings are q and -q. Find the potential difference between the centres of the rings separated by a distance a. |
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| 676 | There is an infinitely long straight thread carrying a charge with linear density λ = 0.40 μC/m. Calculate the potential difference between points 1 and 2 if point 2 is removed η = 2.0 times farther from the thread than point 1. |
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| 677 | Find the electric field potential and strength at the centre of a hemisphere of radius R charged uniformly with the surface density σ. |
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| 678 | A very thin round plate of radius R carrying a uniform surface charge density σ is located in vacuum. Find the electric field potential and strength along the plate's axis as a function of a distance l from its centre. Investigate the obtained expression at l → 0 and l >> R. |
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| 679 | Find the potential φ at the edge of a thin disc of radius R carrying the uniformly distributed charge with surface density σ. |
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| 680 | Find the electric field strength vector if the potential of this field has the form φ = ar, where a is a constant vector, and r is the radius vector of a point of the field. |
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