1. Formulate and write mathematically the four Maxwell's equations in integral form. 2. Explain the relation between electric and magnetic fields. Why electromagnetic waves do propagate in space? 3. How has Maxwell introduced the concept of a ""displacement current""?
Answers
1) The four Maxwell's Equations in the integral form are as follows:
a) Gauss's law for electric fields: ∮E⋅dA = Q/ε₀, where E is the electric field, dA is an infinitesimal area element, Q is the total charge enclosed by the surface, and ε₀ is the permittivity of free space.
b) Gauss's law for magnetic fields: ∮B⋅dA = 0, where B is the magnetic field and dA is an infinitesimal area element.
c) Faraday's law of electromagnetic induction: ∮E⋅dl = -dΦ/dt, where dl is an infinitesimal length element, Φ is the magnetic flux through a surface bounded by the closed loop, and t is time.
d) Ampere's law with Maxwell's correction: ∮B⋅dl = μ₀(I + ε₀(dΦ_E/dt)), where B is the magnetic field, dl is an infinitesimal length element, I is the total current passing through the surface bounded by the closed loop, μ₀ is the permeability of free space, ε₀ is the permittivity of free space, and dΦ_E/dt is the time rate of change of electric flux through the surface bounded by the closed loop.
2) The relationship between Electric and Magnetic Fields:
Electric fields are created by charged particles and exert a force on other charged particles within their vicinity. Magnetic fields, on the other hand, are created by moving charged particles and exert a force on other moving charged particles.
The two fields are related in that a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. This relationship is described by Maxwell's equations, which predict the existence of electromagnetic waves.
Propagation of Electromagnetic Waves:
Electromagnetic waves propagate in space because they are self-sustaining disturbances that do not require a medium to travel through.
They consist of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of wave propagation. These waves can be generated by accelerating charges or by changing magnetic fields, and they travel at the speed of light.
3) The Maxwell introduced the concept of a Displacement Current:
Maxwell introduced the concept of a displacement current to explain how changing electric fields can produce magnetic fields. He realized that a time-varying electric field could create a changing electric flux, which in turn could produce a magnetic field.
This changing electric flux is equivalent to a current, even though no charges are actually moving. Maxwell called this current the displacement current, and he added it to Ampere's law to obtain a more complete description of electromagnetic phenomena.
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Related Questions
Given the equation of a line: 6x-3y+8=0,
Rearrange the equation in slope y-intercept form. [2]
Answers
Answer: y = 2x + 8/3
Step-by-step explanation:
6x - 3y +8 = 0
-3y + 8 = -6x
-3y = -6x - 8
y = 2x + 8/3
Answer:
y = 2x + 8/3
Step-by-step explanation:
slope y-intercept form => y = mx + b
=> -3y = -6x - 8
=> y = 2x + 8/3
What is the slope of the line through (0,-1) and (2,3)?
Answers
(3+1)/(2-0) = 4/2 = 2
The slope is 2
In each of problems 5 through 11, find the general solution of the given differential equation
Answers
The complete question is
"Find the general solution of the given differential equation
y''-y=0, y1(t)=e^t , y2(t)=cosht
The function \(y(t)=e^t\) is the solution of the given differential equation.
The function y(t)=cosht is the solution of given differential equation.
What is a function?
The function is a type of relation, or rule, that maps one input to specific single output.
Given;
\(y_1(t) = e^t\)
Given differential equations are,
y''-y = 0
So that,
\(y' (t) = e^t, y'' (t) = e^t\)
Substitute values in the given differential equation.
\(e^t -e^t=0\)
Therefore, the function \(y(t)=e^t\) is the solution of the given differential equation.
Another function;
\(y(t)=cosht\)
So that,
\(y"(t)=sinht\\\\y"(t)=cosht\)
Hence, function y(t)=cosht is solution of given differential equation.
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Two angles are supplementary. One angle is twenty less than four times the other angle. Find the measure of the larger angle.
180°
140°
130°
40°
NO LINKS
Answers
If two angles are supplementary, then they add up to 180.
Angle 1 = twenty less than four times the other angle
Angle 2 = the other angle
Angle 2 = 1x
Angle 1 = 4x - 20
Next, we can add up angles 1 and 2 and set the sum equal to 180.
x + 4x - 20 = 180
5x - 20 = 180
5x = 200
x = 40
Now that we know x, we can find the values of our angles.
Angle 1 = 4(40) - 20 = 160 - 20 = 140 degrees
Angle 2 = 40 degrees
Hope this helps!! :)
The graph of a proportional relationship is shown. Describe the real world situation that could be represented by the graph. Be sure to include the meaning of the constant of proportionality
Answers
The real world situation that could be represented by a graph is Given that y varies proportionally with x, with a constant of proportionality k=13 , find y when x=12.
How to illustrate the information?The graph of the proportional relationship equation is a straight line through the origin.
The real-world situation that could be represented by a graph is Given that y varies proportionally with x , with a constant of proportionality k=13 , find y when x=12.
The constant of proportionality is the ratio of two proportional values at a constant value. Two variable values have a proportional relationship when either their ratio or their product gives a constant
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The graph of h(x) = x² was transformed to create the graph of k(x) = -1/7x²? Which of these describes the transformation from the graph of h to the graph of k?
A.) A reflection over the x-axls and a vertical stretch
B.) A reflection over the x-axis and a vertical compression
C.) A reflection over the y-axis and a vertical compression
D.) A reflection over the y-axis and a vertical stretch
Answers
Given:
The functions are:
\(h(x)=x^2\)
\(k(x)=-\dfrac{1}{7}x^2\)
To find:
The transformation from the graph of h to the graph of k.
Solution:
The transformation is defined as
\(k(x)=ah(x+a)=\) .... (1)
Where, a is stretch factor
If 0<|a|<1, then the graph compressed vertically by factor |a| and if |a|>1, then the graph stretch vertically by factor |a|.
If \(a<0\), then the graph of h(x) reflected across the x-axis.
We have,
\(h(x)=x^2\)
\(k(x)=-\dfrac{1}{7}x^2\)
It can be written as
\(k(x)=-\dfrac{1}{7}h(x)\) ...(2)
On comparing (1) and (2), we get
\(a=-\dfrac{1}{7}<0\)
The graph of h(x) reflected across the x-axis.
\(|a|=\dfrac{1}{7}<1\)
So, the graph is compressed vertically.
It means the graph of h reflection over the x-axls and a vertical stretch to get the graph of k.
Therefore, the correct option is B.
For questions 1 - 8, find f-'(x), the inverse of f(x). 1. f(x)= x + 12 2. f(x)= -2 3. f(x)=2x +12 4. f(x)= 3x -4 5. f(x)= 2x + 3 6 6. f(x)=
Answers
Answer:
1) inverse: x - 12
2) inverse: 3x + 6
3) inverse: x/2 - 6
4) inverse: x/3 + 4/3
5) inverse: 3x - 3/2
6) inverse: -3x/2 + 15/2
7) inverse: x/5 - 3
8) inverse: 2x/5 + 1/5
An ant can crawl 4/5 yard in 1 minute. How long will it take for it to crawl 12 yards?
Answers
Answer:
15 minutes
Step-by-step explanation:
Divide 12 yards by 4/5 yards.
12 ÷ 4/5 =
12 * 5/4 =
15
Thus, it would take the ant 15 minutes to crawl 12 yards.
Give me some tips for the Shsat that is tomorrow.
Answers
Answer:
1. Sleep well
2. Eat a proper meal to increase your energy
3. Do the ones you are good at first. Skip the harder questions
4. There are 2 parts. One is ELA while the other is math. Do the one that you are good at first so you wouldn't be annoyed.
5. When you are finished filling out the test slip double-check your work that way you could see which one you got wrong.
6. Have faith in yourself. :)
Step-by-step explanation:
motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production process. assume a production process produces items with a mean weight of 11 ounces. a. the process standard deviation is , and the process control is set at plus or minus standard deviations. units with weights less than or greater than ounces will be classified as defects. what is the probability of a defect (to 4 decimals)? 0.3173 in a production run of parts, how many defects would be found (to the nearest whole number)? 16 b. through process design improvements, the process standard deviation can be reduced to . assume the process control remains the same, with weights less than or greater than ounces being classified as defects. what is the probability of a defect (to 4 decimals)? in a production run of parts, how many defects would be found (to the nearest whole number)? c. what is the advantage of reducing process variation, thereby causing a problem limits to be at a greater number of standard deviations from the mean?
Answers
(a) (i) 0.3174 = 31.74% probability of a defect
(ii) The expected number of defects for a 1,000-unit production run is 317.
(b) (i) 0.0026 = 0.26% probability of a defect
(ii) The expected number of defects for a 1,000-unit production run is 3.
(C) Reduces the process standard deviation and causes no change in the number of defects.
When the distribution is normal, we use the z-score formula.
In a set with mean and standard deviation, the score of a measure X is given by:
Z = X-μ /σ
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Question a:
We have that: μ = 10 and σ = 0.12
(i). Calculate the probability of a defect.
Less than 9.88 or greater than 10.12. These probabilities are equal, so we find one and multiply by 2.
Probability of less than 9.88:
This is the p-value of Z when X = 9.88. Thus,
Z = X-μ /σ
⇒ Z = 9.88 - 10/0.12
⇒ Z = -1, has a p-value of 0.1587
⇒ 2× 0.1587 = 0.3174
This means 0.3174 = 31.74% probability of a defect
(ii) Calculate the expected number of defects for a 1,000-unit production run.
The expected number of defects is 31.74% of 1000. So
0.3174*1000 = 317.4
Rounding to the nearest integer
The expected number of defects for a 1,000-unit production run is 317.
Question (b):
The mean remains the same, but the standard deviation is now
(i) Calculate the probability of a defect.
Less than 9.88 or greater than 10.12. These probabilities are equal, so we find one and multiply by 2.
Probability of less than 9.88:
This is the p-value of Z when X = 9.88. Thus,
Z = X-μ /σ
⇒ Z = 9.88 -10/0.04
⇒ Z = -3
⇒ Z = -3, has a p-value of 0.0013
which means 2× 0.0013 = 0.0026
from which 0.0026 = 0.26% probability of a defect
(ii) Calculate the expected number of defects for a 1,000-unit production run. The expected number of defects is 31.74% of 1000. Thus,
⇒ 0.0026*1000 = 2.6
Rounding to the nearest integer
The expected number of defects for a 1,000-unit production run is 3.
(C) The advantage of reducing process variation, thereby causing problem limits to be at a greater number of standard deviations from the mean Reduces the process standard deviation and causes no change in the number of defects.
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If the correlation between variables is 0.70, what percent of the variance is shared variance? a. 70%. b. 30%. c. 49%. d. 51%.
Answers
The correct answer is c. 49%.
To calculate the shared variance, we need to square the correlation coefficient.
0.70 squared = 0.49
This means that 49% of the variance is shared variance between the two variables. Therefore, option c is the correct answer.
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If the correlation between variables is 0.70, what percent of the variance is shared varianceis c. 49%.
When the correlation between two variables is 0.70, it means that 49% of the variance is shared variance. This is because the correlation coefficient represents the strength and direction of the linear relationship between two variables, and the coefficient of edterminationd (r-squared) is the proportion of the variance in one variable that can be explained by the variance in the other variable. In this case, the r-squared is 0.70^2, which equals 0.49 or 49%.
Therefore, the answer to the question is that 49% of the variance is shared variance when the correlation between variables is 0.70.
The correct option is c. 49%.
To find the shared variance (also known as the coefficient of determination) between two variables, you need to square the correlation coefficient. In this case, the correlation coefficient is 0.70. When you square 0.70 (0.70 * 0.70), you get 0.49, which represents 49% shared variance.
If the correlation between variables is 0.70, the percent of the variance that is shared variance is 49%.
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If line HC and line GD are parallel, which statement is true?
Answers
Answer:
the answer is d
Step-by-step explanation:
hopefully this helps :)
Triangle BCD, with vertices B(4,-7), C(6,-8), and D(7,-2), is drawn on the coordinate
grid below.
S
Answer: A =
6
7
D
9
What is the area, in square units, of triangle BCD?
units
Submit Answer
K
Answers
Answer: The area is 6.5
(Quick I'm being timed)
Find the value of the variable using these steps.
0.4x + 3.9 = 5.78
1. Subtraction property of equality
2. Division property of equality
X = ____
Answers
Answer:
x = 4.7
Step-by-step explanation:
You will need to subtract 3.9 from 5.78, then divide by 0.4. The answer is 4.7 because you are left with x = 4.7.
Hope this helps you!
help with all pleas
1. what is 3% of 84
2. 378 is what percent of 450
3. 8 is what percent of 128
Answers
Answer:
Hello There!!
Step-by-step explanation:
1. 2.52
2. 84
3. 6.25
hope this helps,have a great day!!
~Pinky~
hi can you please help me
Answers
Answer:
See attached image
Step-by-step explanation:
I'm uncertain as to the "best statement." None seem to accurately describe the situation. I picked the one that was least incorrect (if that makes any sense).
Jaguars
3 14 7
7 21 10
17 6 14
24 21 7
10 17 3
Eagles
3 24 14
27 10 13
10 21 24
17 27 7
40 37 55
Which team had the best overall record for the season? Determine the best measure of center to compare, and explain your answer
O Eagles; they have a larger median value of 21 points
O Jaguars, they have a larger median value of 10 points
O Eagles, they have a larger mean value of about 22 points
OJaguars, they have a larger mean value of about 12.1 points
Answers
A team that had the best overall record for the season and the best measure of center to compare include the following: C. Eagles; they have a larger mean value of about 22 points.
From the data set for Jaguars, we have:
3, 14, 7, 7, 21, 10, 17, 6, 14, 24, 21, 7, 10, 17, 3
Mean of Jaguars = [3 +14 +7 +7 +21+ 10 +17 +6+ 14+ 24+ 21+ 7+ 10+ 17 +3]/15
Mean of Jaguars = 181/15
Mean of Jaguars = 12.07
Median of Jaguars = 10.
From the data set for Eagles, we have:
3, 24, 14, 27, 10, 13, 10, 21, 24, 17, 27, 7, 40, 37, 55
Mean of Eagles = [3 +24+ 14 +27 +10+ 13+ 10 +21 +24+ 17+ 27+ 7+ 40+ 37 +55]/15
Mean of Eagles = 329/15
Mean of Eagles = 21.93 ≈ 22.
The median of Eagles = 21.
Since Eagles have both a larger mean and median value, the best measure of center to compare would is the mean.
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Which is the best estimate of the difference between 6 7 8 678 and 2 1 8 218
Answers
Answer: C.5
Step-by-step explanation:
What number Am I?
Elmo wants answers
Answers
Answer:
20
Step-by-step explanation:
Since it's a multiple of 5 between 18 and 30, possible answers are:
20 and 25
The GCF of 18, 36, and the number is 2, and 25 is not divisible by 2, leaving only 20.
i need help in ap calc ab pls hurryyyyy
Answers
Answer:
a. (-pi/2, pi/2)
b. (-pi, pi/2) U (pi/2, pi)
c. -pi/2
d. pi/2
Step-by-step explanation:
g'(x)=cos(t) (-pi,pi) Fundamental Theorem of Calculus
It increases while g'(x) is positive, so (-pi/2, pi/2)
It decreases while g'(x) is negative so (-pi, pi/2) U (pi/2, pi)
The local minimum occurs when g'(x) switches from negative to positive
The local maximum occurs when g'(x) switches from positive to negative
City A is 10 miles from city B and 5 miles from city C. Cities A, B, and C form a right triangle at A. A road connects cities B and C directly. Find the length of this road.
Answers
Answer:
11.18 miles
Step-by-step explanation:
The sides of a right triangle a, b, c are related by the equation
a^2 + b^2 = c^2
where c is the longest side of the triangle
Hence given that A connects B and C directly and forms a right angle at A
5^2 + 10^2 = c^2
Where c is the length of the road
25 + 100 = c^2
125 = c^2
c = 125^1/2
= 11.18 miles
use properties of operations to complete the equivalent expressions
2(r + 3)
Answers
Which relation is also a function?
Answers
Answer:
Hi san, A relation from a set X to a set Y is called a function if each element of X is related to exactly one element in Y. That is, given an element x in X, there is only one element in Y that x is related to. For example, consider the following sets X and Y.
This should help you a bit.
a convention manager finds that she has $1320, made up of twenties and fifties. she has a total of 48 bills. how many fifty-dollar bills does the manager have?
Answers
The required manager has 12 fifty-dollar bills as of the given condition.
Let's denote the number of twenty-dollar bills as "x" and the number of fifty-dollar bills as "y".
We know that the convention manager has a total of 48 bills, so:
x + y = 48
We also know that the total amount of money she has is $1320, which can be expressed as:
20x + 50y = 1320
To solve for "y", we can rearrange the first equation to get:
y = 48 - x
Then substitute this expression for "y" in the second equation:
20x + 50(48 - x) = 1320
Expanding the expression and simplifying:
20x + 2400 - 50x = 1320
-30x = -1080
x = 36
So the manager has 36 twenty-dollar bills. To find the number of fifty-dollar bills, we can use the first equation:
x + y = 48
36 + y = 48
y = 12
Therefore, the manager has 12 fifty-dollar bills.
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Which statement(s) is(are) true about cones? Statement 1: A cone has a triangular base. Statement 2: A cone is 3 times larger than a cylinder with the same height and radius. Statement 3: A cone is one-third the size of a cylinder with the same height and radius. Statement 4: A cone has a circular base.
Answers
Answer:
Statement 3: A cone is one-third the size of a cylinder with the same height and radius.
Statement 4: A cone has a circular base.
Step-by-step explanation:
Statement 3: A cone is one-third the size of a cylinder with the same height and radius.
Statement 4: A cone has a circular base.
The true statement is:
Statement 3: A cone is one-third the size of a cylinder with the same height and radius.
Statement 4: A cone has a circular base.
What is Cone?A cone is a three-dimensional solid geometric shape having a circular base and a pointed edge at the top . A cone has one face . There are no edges for a cone.
Properties of Cone
A cone is a shape that has a curved surface and a circular base. The following properties of a cone help us identify it easily. They are as follows.A base of a cone is circular.There is one face, one vertices, and no edges for a cone.The slant height of a cone is the length of the line segment joining the of the cone to any point on the circumference of the base of the cone.A cone have circular base at a perpendicular distance is called a right circular cone.A cone have circular base is an oblique cone.As, from the properties of cone we can say that.
A cone is one-third the size of a cylinder with the same height and radius. and, : A cone has a circular base..
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A newspaper surveyed households to determine how likely they were to subscribe to their publication. The results are shown below by annual household income.
Household Income
< $60,000 > $60,000
I would never purchase a newspaper subscription. 40 63
I might or might not purchase a newspaper subscription. 53 70
I would probably purchase a newspaper subscription. 65 32
I already purchase a newspaper subscription. 73 16
Identify the percentage of families with an income more than $60,000 who might or might not purchase a newspaper subscription. What does this number represent?
Answers
Complete Question
The complete question is shown on the first uploaded image
Answer:
The correct option is C
Step-by-step explanation:
The sample
Household Income
< $60,000 > $60,000
I would never purchase a newspaper subscription. 40 63
I might or might not purchase a newspaper subscription. 53 70
I would probably purchase a newspaper subscription. 65 32
I already purchase a newspaper subscription. 73 16
Generally the sample size is mathematically represented as
\(n = 40 + 53 + 65+ 73 + 63 + 70 + 32+ 16\)
=> \(n = 412\)
Generally the number of the sample with an income greater $60,000 is mathematically represented as
\(N = 63 + 70 + 32 + 16\)
=> \(N = 181\)
Generally the number of families with an income more than $60,000 who might or might not purchase a newspaper subscription is
\(X = 70\)
Generally the percentage of families with an income more than $60,000 who might or might not purchase a newspaper subscription is
\(P(X) = \frac{70}{181} * 100\)
=> \(P(X) = 39\%\)
4. fifty people purchase raffle tickets. three winning tickets are selected at random. if each prize is $500, in how many different ways can the prizes be awarded?
Answers
As per the concept of combination method, there are 19,600 different ways in which the prizes can be awarded.
To solve this problem, we can use a combination formula. A combination is a selection of objects without regard to order. In other words, it doesn't matter which order the winning tickets are selected in - we just want to know how many different combinations of three tickets can be selected from fifty.
The combination formula is ⁿCₓ, where n is the total number of objects to choose from, and x is the number of objects to be chosen. In this case, n = 50 (the total number of tickets), and r = 3 (the number of winning tickets).
So, we can calculate the number of ways the prizes can be awarded using the following formula:
⁵⁰C₃ = 50! / (3! * (50-3)!)
Simplifying this equation, we get:
⁵⁰C₃ = (504948) / (321)
⁵⁰C₃ = 19,600
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Factor completely.
4x² - 4x + 1
Answers
whose perfect square should be added to the sum of the greatest three-digit prime number and the smallest two-digit prime number, so that the resultant number is perfect square of 32?
Answers
The greatest 3 digit prime number is 997 and the smallest 2 digit prime number is 11. 997+11 = 1008
32^2 = 1024
1024-1008 = 16.
root(16) = 4
student researchers wanted to see whether a short delay between seeing a list of words and when people were asked to recall them would hinder memorization. the subjects were shown a list of words to memorize for 1 minute and were then given 1 minute to recall as many words as they could. each subject did this once with no delay between memorizing and recall and another time with a 30-second wait between memorizing and recall. they were randomly assigned the order of the two conditions. the number of words memorized under each condition can be found in the statcrunch data set memorizingwords. questions: use the appropriate simulation applet to calculate an approximate p-value. is there strong evidence that a short delay hinders the memorization process? explain.
Answers
The appropriate simulation applet to calculate an approximate p-value is 0.0077.
The strong evidence that a short delay hinders the memorization process is strong data suggests that a brief delay impairs memorization.
Delay or no delay, answer: the number of words remembered
The mean distance between words remembered with and without delay over time is null, or zero.
Alt: The mean distance in words remembered over the long term (with no delay minus with delay) is higher than 0.
Yes, the majority of the lines do skew to the right. The mean for no delay is 10.55 whereas that for with delay is 8.7.
The mean distance in words remembered is 1.85.
1)0.0077 as the p-value
2)Strong data suggests that a brief delay impairs memorization.
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