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Carlos Nava

Carlos N.

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A sphere of radius $R$ carries a polarization $$\mathbf{P}(\mathbf{r})=k \mathbf{r},$$ where $k$ is a constant and $\mathbf{r}$ is the vector from the center. (a) Calculate the bound charges $\sigma_{b}$ and $\rho_{b}$. (b) Find the field inside and outside the sphere.

A sphere of radius $R$ carries a polarization $$\mathbf{P}(\mathbf{r})=k \mathbf{r},$$ where $k$ is a constant and $\mathbf{r}$ is the vector from the center. (a) Calculate the bound charges $\sigma_{b}$ and $\rho_{b}$. (b) Find the field inside and outside the sphere.

Introduction to Electrodynamics

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Arun Bana verified

Numerade educator

A coin is placed at the bottom of a water reservoir (n = 1.33) 1 m deep. On the floating water a 20 cm thick layer of mineral oil (n = 1.48). The medium after mineral oil is air (n = 1). How far from the mineral oil-air separation surface does the coin appear when viewed in the normal direction?

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A thick lens is placed at the end of a fish tank containing a Transparent liquid whose refractive index is 1.420. The lens has surfaces of radii of curvature of R1 = +3.80 cm and R2 = ?1.90 cm, a thickness of 4.60 cm and is made of a glass whose index of refraction is 1.620. If R2 is in contact with the liquid, obtain (a) the front and back focal lengths, (b) the power of the lens. (c) Obtain also the distances from the vertices to the foci and to the respective main planes.

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Khoobchandra Agrawal verified

Numerade educator

Show that the effective focal length of a sphere of radius R made of a material whose refractive index is n and surrounded by air is f = nR / (2 (n - 1)). Show that the two principal planes coincide and cut the sphere in half.

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A thick lens is placed at the end of a fish tank containing a Transparent liquid whose refractive index is 1.420. The lens has surfaces of radii of curvature of R1 = +3.80 cm and R2 = ?1.90 cm, a thickness of 4.60 cm and is made of a glass whose index of refraction is 1.620. If R2 is in contact with the liquid, obtain (a) the front and back focal lengths, (b) the power of the lens. (c) Obtain also the distances from the vertices to the foci and to the respective main planes.

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Vishal Gupta verified

Numerade educator

An equiconvex lens 3.50 cm thick has radii of curvature of 5.20 cm and is made of a glass with refractive index n = 1.680. Calculate (a) the focal length of the lens and (b) the power of the lens. Obtain the positions of the principal planes and the distances of the vertices to the respective foci.

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5) Let f (z) = (1 + i) ((z - (5 + 2i)) ^ 2 + 1 - 6i). What is the direct image of: (a) the two horizontal rays and the two vertical rays emanating from the point 5 + 2i? (b) the circle with center at 5 + 2i and radius 1?

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1) Let a be a nonzero complex number. Prove that f (z) = Exp (az) is a periodic function, and find its period. 2) Separate the function: Exp (1 / z) into its real and imaginary parts. 3) Identify which points must be excluded from the complex plane for the function f (z) = Log (z ^ 3 - 1) to be continuous. 4) Identify which points must be excluded from the complex plane so that the function f (z) = (?z ^ 2 + 1) is continuous. 5) Let f (z) = (1 + i) ((z - (5 + 2i)) ^ 2 + 1 - 6i). What is the direct image of: (a) the two horizontal rays and the two vertical rays emanating from the point 5 + 2i? (b) the circle with center at 5 + 2i and radius 1?

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A hemispherical glass of refractive index 1.5 and radius R is placed on a flat mirror (see the figure). A small object of height h is located on the axis of the hemisphere at a distance 2R from the vertex. Find the position of the first image formed by the refraction on the spherical surface. Where is the image reflected by the mirror formed? Where is the final image formed by the entire optical system located? How big is the final image? Indicate if this image is inverted or not.

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417 / 5000 Resultants de traducción The left end of a glass rod with refractive index n = 1.5 is a convex surface of radius equal to 2 cm. If the bar is in air, find the two focal points and make a diagram of the optical system. If we place an arrow of 1 cm at 90 ° with respect to the axis of the bar and 8 cm to the left of the vertex, find the position of the image of the arrow. Make a scale drawing of the system.

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A small fish is at point O (see the figure) 7.5 cm from the center C of a spherical fish tank 30 cm in diameter. Find the position and lateral magnification of the image seen by the observer in E. Find the position and lateral magnification of the image of the observer's eye seen by the fish if the observer is 30 cm from the fish tank.

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