6. A long core-shell structure is made to transmit an optical wave through total internal reflection within the core region. If the refractive indices for the core and the shell are respectively \(n_1 = 1.5\) and \(n_2 = 3.0\), as depicted in the figure. What is the minimum value of the angle \(\theta\) for a successful transmission?

(A)30°

(B)45°

(C)60°

(D)90°

(E)None of the above

1. In free space, there are two point charges of Q\(_1\) = 2q and Q\(_2\) = 6q located at (0, 2r, r) and (r, 0, 3r) in rectangular (Cartesian) coordinate system. What is the magnitude of the force experienced by Q\(_2\)? (A)\(\frac{q^2}{2\pi\varepsilon_0 r^2}\) (B)\(\frac{3q^2}{4\pi\varepsilon_0 r^2}\) (C)\(\frac{q^2}{3\pi\varepsilon_0 r^2}\) (D)\(\frac{2q^2}{3\pi\varepsilon_0 r^2}\) (E)None of the above

2. A sphere with radius a contains a uniform volume charge density of \(\rho\) in the region: 0 ＜ r ＜ a. If the sphere is isolated by a conducting spherical shell with inner radius b and outer radius c placed concentrically, as shown in figure. What is the electric field intensity in the region b ＜ r ＜ c? (A)\(\frac{\rho a^3}{3\varepsilon_0 r^2}\) (B)\(\frac{\rho r}{3\varepsilon_0}\) (C)0 (D)\(\frac{\rho a^3}{3\varepsilon_0 b^2}\) (E)None of the above

3. A straight wire carrying a current I in the z direction is located along the z axis in free space. If the wire is infinitely long and infinitely thin, what is the magnetic field intensity at the point (0, L, 0) in rectangular coordinate system? (A)\(\frac{I}{2\pi L}\hat{x}\) (B)\(\frac{I}{2\pi L}\hat{z}\) (C)\(\frac{I}{2\pi L}(-\hat{x})\) (D)\(\frac{I}{4\pi L}\hat{x}\) (E)None of the above

4. In a perfect dielectric medium with the permeability of \(\mu\), there exists an electric field given as \(\mathbf{E} = E_0\cos(\omega t - kz)\hat{a}_x\), where \(E_0\) is the peak value, \(\omega\) is the frequency, k is a constant, and \(\hat{a}_x\) is the unit vector in x direction. What is the magnetic field (H) in the region? (A)\(\frac{kE_0}{\mu\omega}\cos(\omega t - kz)\hat{a}_y\) (B)\(\frac{kE_0}{\mu\omega}\cos(\omega t - kz)(-\hat{a}_y)\) (C)\(\frac{\mu\omega E_0}{k}\cos(\omega t - kz)\hat{a}_y\) (D)\(\frac{kE_0}{\mu\omega}\sin(\omega t - kz)\hat{a}_y\) (E)None of the above

5. A uniform electromagnetic plane wave is traveling in free space and its electric field is given by \(\mathbf{E} = 5\cos(\omega t - 2z)\hat{a}_y\) V/m, what is the angular frequency, \(\omega\) in (rad/s), of the plane wave? (A)\(3\times 10^8\) (B)\(4\times 10^8\) (C)\(5\times 10^8\) (D)\(6\times 10^8\) (E)None of the above

7. An object is placed at the position being 20 cm from a convex lens with the focal length of 10 cm. What is the distance between the image and the lens? (A)20 cm (B)10 cm (C)40 cm (D)30 cm (E)None of the above

8. The spherical waves of wavelength \(\lambda\) emitted from a single point source are passing through two small holes, S\(_1\) and S\(_2\), which are separated by the distance of a, as shown in the figure. If c ＞＞ a, what is the approximated condition for destructive interference at a point P on the screen? (A)\(a\sin\theta = n\lambda\) (B)\(a\sin\theta = (n + \frac{1}{2})\lambda\) (C)\(a\sin\theta = \frac{n\lambda}{2}\) (D)\(2a\sin\theta = n\lambda\) (E)None of the above

9. A laser beam of wavelength \(\lambda\) shined on a spectrometer with the grating spacing a is split into two beams of wavelengths \(\lambda_1\) and \(\lambda_2\) due to the 2nd order diffraction. If \(\lambda_2\) is slightly longer than \(\lambda_1\) and \(\theta\) is the diffraction angle for \(\lambda_1\), what is the difference in diffraction angles of the two beams? (A)\(\frac{\lambda_2 - \lambda_1}{2a\cos\theta}\) (B)\(\frac{\lambda_2 - \lambda_1}{a\cos\theta}\) (C)\(\frac{2(\lambda_2 - \lambda_1)}{a\cos\theta}\) (D)\(\frac{\lambda_2 - \lambda_1}{a\sin\theta}\) (E)None of the above

10. A light beam is traveling in a structure with three layers of different refractive indices: \(n_1 = 2\), \(n_2 = \sqrt{2}\), \(n_3 = 1\). If the beam impinges at the interface between \(n_1\) and \(n_2\) with the incident angle of 60°, as shown in the figure, what is \(\theta\) when the light beam enters the layer with \(n_3 = 1\)? (A)30° (B)45° (C)60° (D)90° (E)None of the above

11. A particle is strictly confined in a region of length L, but moving freely in the region. Two possible wavefunctions, \(\phi_1\) and \(\phi_2\), of the particle are those shown in the figure. When the particle is in the state corresponding to the wavefunction \(\phi_1\), its total energy is 1 eV. What is its total energy in the state corresponding to \(\phi_2\)? (A)1 eV (B)2 eV (C)3 eV (D)4 eV (E)None of the above

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