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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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|...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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...
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...
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...