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  1. C

    Amplitude of oscillations

    Homework Statement A little amount of sand is spilt over horizontal membrane that oscillates with frequency f=500Hz in vertical plane. If sand grains are jumping to the height h=3mm with respect to the equilibrium position, find amplitude of oscillation of membrane. Homework Equations ω=2πf...
  2. C

    Frequency of wire oscillations

    Homework Statement There's a horizontal thin wire whose mass is negligible and whose length is l=1m. It is strained with constant force F=10N. If we place a tiny pellet in the middle of wire (mass of pellet is m=1g) and then we bring wire out of equilibrium position (moving out of it's original...
  3. C

    Total magnetic moment of simple system.

    Homework Statement If there's a current I flowing through the rectangular conductor and it's located in the magnetic field in such way that normal vector of the surface that this rectangle forms closes the angle of 60 degrees with magnetic induction vector find total magnetic moment of this...
  4. C

    Two coils of wire with the same size and shape

    Homework Statement Two coils of wire that have same shape and dimensions are thickly rolled up so that the coincide, they only differ in number of coils (windings) N1 for the firs one and N2 for the second one (N1<N2). When there's constant current in the first one and there's no current in the...
  5. C

    Magnetic field of a ferromagnetic cylinder

    Homework Statement We have a very long ferromagnetic cylinder with square cross section, with side length a. Cylinder is homogeneosly magnetized over his volume so that the magnetization vector is parallel to the axis of cylinder. Cylinder is in vacuum. Find the current appeared in the cylinder...
  6. C

    Force acting on conducting contour

    Homework Statement We have contours C1 and C2 located in vacuum and there are constant currents in them, I1 and I2 respectively. Find the expression for magnetic force that acts on one very small element (dl) of the circle C2 and it's coming from one very small element of the contour C1...
  7. C

    Superconductive contour

    1. The problem statement, all variables and given/known data We have a superconductive contour in the shape of circle with radius ##a##. Inductance of contour is ##L##, when the contour is out of magnetic field, there's no current in it. What's the current in the contour when constant magnetic...
  8. C

    Force and energy of the magnet

    Homework Statement If we have a system that is shown in the picture, the upper piece is magnet, and if we know the following data: S = 1cm2 which is area of material, S0 = 1.2cm2 which is the area of fissure. To bring these tow pieces of this system magnetic induction in the fissure should be...
  9. C

    Limit of series.

    Homework Statement If we have a number sequence such that: a0, a1 are given, and every other element is given as ##a_n=\frac{(a_{n-1} + a_{n-2})}{2} then express an in terms of a0, a1 and n , and fin the limit of an Homework Equations The Attempt at a Solution If i try to express a3 in terms...
  10. C

    Determinant nxn.

    Homework Statement I have to solve the following determinant ## D_n=\begin{vmatrix} 1 & 1 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & 1 & \cdots & 1 & 2 & 1 \\ 1 & 1 & 1 & \cdots & 2 & 1 & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 1 & 2 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & 1 &...
  11. C

    Finding complex number with the lowest argument.

    Homework Statement Of all complex numbers that fit requirement: ## |z-25i| \leq 15## find the one with the lowest argument. Homework Equations The Attempt at a Solution z=a + ib (a, b are real numbers) ## \sqrt{a^2 + (b-25)^2} \leq 15 \\ a^2 + (b-25)^2 \leq 225 ## The lowest possible...
  12. C

    Equation with complex number

    Homework Statement Solve the following equation: ## (1+a)^n=(1-a)^n## where a is complex number and n is natural number Homework Equations Euler's formula The Attempt at a Solution I've tried something like this ## (1+a)^n=(1-a)^n \\ (\frac{1+a}{1-a})^n=1 ## But i really have no idea...
  13. C

    Complex polynomial

    Homework Statement It is known that roots of complex polynomial: ##P_n (z) = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0## are the following complex numbers: ##\alpha_1, \alpha_2, \cdots, \alpha_n ## Find the product: ##\prod = (\alpha_1 + 1)(\alpha_2 + 1)\cdots(\alpha_n +1)## Homework...
  14. C

    Sets of matrices...

    Homework Statement If there are two sets of matrices ##S = \begin{Bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} | a, b, c, d \in \mathbb{C} \end{Bmatrix} ## and ##M = \begin{Bmatrix} \begin{bmatrix} a & b \\ -\overline{b} & \overline{a} \end{bmatrix} | a, b \in \mathbb{C} \wedge |a|...
  15. C

    Cosine and sine of 2Pi/5 problem

    We have a complex number ω = cos(2π/5) +isin(2π/5) and we have two complex numbers a and b, such that: a= ω + ω4 and b= ω2 + ω3. I have to prove that a + b = -1 and a*b= -1. Then, based on that, determine cos(2π/5) and sin(2π/5). I've tried to solve this using trigonometry. First a+b, i got...
  16. C

    Equation with complex numbers

    I have to solve the following equation: z4=i*(z-2i)4 Now, i tried to move everything but i (imaginary number) to the left side and then find the 4-th root of i, when i did that, i had four solutions, with one of them being eiπ/8. But i don't know what to do with the left side, since i get way...
  17. C

    Solve equation with unknown x

    Homework Statement I have to solve the following equation: Homework Equations The Attempt at a Solution I know that since the right side is 1 and on the left side i have i (imaginary number) it means that i could rewrite right side as cos0 + isin0 since it's the same, but what can i do...
  18. C

    Finding solution of equation in complex domain?

    (1+a)n = (1-a)n I tried following: (1+a)n = (1-a)n [(1+a)/(1-a)]n=1 but what can i do next?
  19. C

    Factorize polynomial

    I should factorize following polynomial: P(x)=x^2n + 2cos(naπ)x^n + 1 in ℝ if i know that a is irrational number. Things that confuse me here are following: 1. When factorizing polynomials, i have known exponents (unlike here, where i have 2n and n) so i don't know what to do with them? 2...
  20. C

    Magnetic field vector due to linear conductor

    Homework Statement Through linear conductor flows current I, with direction shown in the picture. Axis where conductor is placed is common edge of three areas with different ferromagnetic materials. They form angles θ1, θ2, θ3 (θ1 + θ2 + θ3 = 2π). If space is filled with homogeneous materials...
  21. C

    Proof using mathematical induction

    Prove that n^5 - 5n^3 + 4n is divisible by 120. for every natural number n greater or equal to 3. First, i checked if it works for n=3 and it does, so i could assume it works for some k>=3 so i could write k^5 - 5k^3 + 4k as 120*a a is natural number so for k+1 i have: (k+1)^5 - 5(k+1)^3...
  22. C

    Induced voltage from a rod sliding on rods in a B-Field

    Homework Statement Two long rods with distance between them a=100mm, over those two rods there's another rod sliding, friction between them is negligible. The rods are placed in homogenous magnetic field whose B is equal 1T, with direction shown in the picture, between the rods there is voltage...
  23. C

    Faraday's disc

    Homework Statement Faraday's disk with radius of axle r1 and radius of disc r2 is shown in the picture below. Disc rotates in the homogenous magnetic field whose vector B is vertical to the disc. If disc is spinning 1000 rpm find induced voltage between r1 and r2 and what is maximal value of...
  24. C

    Magnetic induction vector?

    Homework Statement In the system shown in the picture, there's a current whose constant density is J=0,5A/mm^2. System contains two pieces as shown in the picture, in the area where two pieces intersect, there's no current. If R=1mm and a=1,25mm (a - distance between centers of the circles)...
  25. C

    Potential of the point in the origin

    Homework Statement Find the potential of the point in the origin in regard to referring point at infinity if there's charged line laying on x axis as shown in the picture. Homework Equations The Attempt at a Solution I know two ways to find a potential: V=∫dV and V=∫E*dl using...
  26. C

    Maximum voltage on coaxial cable?

    Homework Statement How to determine maximum voltage that coaxial cable, whose height is L and is filled with dielectric in such way that E is the same in the dielectric and in the part of the cable that isn't filled with dielectric (only vacuum),can handle if i have maximum electric field that...
  27. C

    Finding longitudinal force?

    Homework Statement Very long thread, with constant longitudinal charge Q' is placed in a vacuum parallel to a very long conductive strip, whose width is a. thread is placed in the middle of the strip and it's a/2 away from it, if the surface density of charge of the strip is σ, find the...
  28. C

    Electric field and potential at the origin

    Homework Statement I have to find a electric field vector and a potential of the point in the origin of the x0y coordinate system (0,0) due to a longitudinal charge placed on a half circle as shown in the picture, with radius a. Homework Equations Coulombs law. The Attempt at a Solution...
  29. C

    Charge staggered over spherical volume.

    Homework Statement There is charge placed in a volume of a sphere, whose density changes by expression ρ(r)=ρ0a/r for 0<r≤a and ρ(0)=0. Where a and ρ0 are known variables , and r is a distance from the origin. Determine the potential of the point A(0,0,0) with regard to reference point at...
  30. C

    Potential of a point at the z axis of the circular ring?

    So i have a thin circular ring laying at the XoY plane, inner radius of the ring is a, outer is b, density of electricity is given by expression ρ=ρ0*b/r , where ρ0 is a constant and r ∈ (a,b). I have to find a potential of the point P with coordinates (0,0,z). I tried to do it two ways, but...
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