The potential due to a point charge q at the origin may be written as v=q4πϵ0r=q4πϵ0x2+y2+z2√ part a calculate ex using equation ex=−∂v∂x.
Answers
[tex]v= \frac{q}{4 \pi \epsilon _{0} r} = \frac{q}{4 \pi \epsilon _{0} \sqrt{x^{2}+y^{2}+z^{2}} } [/tex]
The x-component of electric field is
[tex]e_{x} = - \frac{\partial v}{\partial x} = -\frac{q}{4 \pi \epsilon {0}}(- \frac{1}{2}) \frac{2x}{(x^{2}+y^{2}+z^{2})^{3/2}} = \frac{qx}{4 \pi \epsilon _{0} (x^{2}+y^{2}+z^{2})^{3/2}} [/tex]
Answer:
[tex]e_{x} = \frac{qx}{4 \pi \epsilon _{0} (x^{2}+y^{2}+z^{2})^{3/2}} [/tex]
Eₓ = [(qx) / [(4πϵ₀r)√(x² + y² + z²)³]
Further explanation
In this problem, we will solve the derivatives of the composite function. The formula used is the chain rule as follows.
[tex]\boxed{ \ If \ f(x) = h(g(x)) \ then \ f'(x) = h'(g(x)) \times g'(x) \ }[/tex]
In words: differentiate the outer function, then multiply it by the derivative of the innermost function.
Given:
[tex]\boxed{ \ V = \frac{q}{4 \pi \epsilon_0 r} \rightarrow V = \frac{q}{4 \pi \epsilon_0 r \sqrt{x^2 + y^2 + z^2}} \ }[/tex]
Question:
Calculate the x-component of electric field by using [tex]\boxed{ \ E_x = - \frac{\delta V}{\delta x} \ }[/tex]
The Process:
This case is a derivative application for electrostatics in physics.
The potential due to a point charge q at the origin may be written as
[tex]\boxed{ \ V = \frac{q}{4 \pi \epsilon_0 r \sqrt{x^2 + y^2 + z^2}} \rightarrow V = \frac{q}{4 \pi \epsilon_0 r}(x^2 + y^2 + z^2)^{-\frac{1}{2}} \ }[/tex]
The outside function is [tex] \boxed{ \ h(x) = \frac{q}{4 \pi \epsilon_0 r}(g(x))^{-\frac{1}{2}} \ } [/tex]The inside function is [tex]\boxed{ \ g(x) = x^2 + y^2 + z^2 \ }[/tex]Let's determine the derivative and run the composite function rule.
The outside function: [tex] \boxed{ \ h'(x) = -\frac{1}{2} \frac{q}{4 \pi \epsilon_0 r}(g(x))^{-\frac{3}{2}} \ } [/tex]The inside function: [tex]\boxed{ \ g'(x) = 2x \ }[/tex]Let us calculate the x-component of electric field by using
[tex]\boxed{ \ E_x = - \frac{\delta V}{\delta x} \ }[/tex]
The composite function rule tells us that
[tex]\boxed{ \ V(x) = h(g(x)) \rightarrow V'(x) = h'(g(x)) \times g'(x) \ }[/tex]
Therefore, [tex]\boxed{ \ E_x = - V'(x) \rightarrow -[h'(g(x)) \times g'(x)] \ }[/tex]
[tex]\boxed{ \ E_x = -[-\frac{1}{2} \frac{q}{4 \pi \epsilon_0 r}(x^2 + y^2 + z^2)^{-\frac{3}{2}} \times 2x] \ }[/tex]
Thus, the result is [tex]\boxed{\boxed{ \ E_x = \frac{qx}{4 \pi \epsilon_0 r}(x^2 + y^2 + z^2)^{-\frac{3}{2}} \ }}[/tex]
[tex]\boxed{\boxed{ \ E_x = \frac{qx}{4 \pi \epsilon_0 r \sqrt{(x^2 + y^2 + z^2)^3} }} \ }}[/tex]
Learn moreUsing the product rule https://brainly.com/question/1578252 The characteristics of electromagnetic waves https://brainly.com/question/727976 Determine the density of our sun at the end of its lifetime https://brainly.com/question/5189537Keywords: calculus, differential, the composite function rule, the potential due to a point charge q, at the origin, physics, application, the chain rule
Related Questions
You took 500 calls in 30 days that means I took a average of
Answers
Answer:
17 calls
Step-by-step explanation:
You took 500 calls in 30 days.
We have to calculate the average calls per day.
Average calls = [tex]\frac{\text{Total calls}}{\text{total days}}[/tex]
= [tex]\frac{500}{30}[/tex]
= 16.7 ≈ 17 calls per day
That means i took an average 17 calls per day.
Tom"s and Tim's ages add up to 21 years. In 3 years, Tom will be twice Tim's age. How old are they now?
Answers
y=Tim´s age.
we suggest this system of equations:
x+y=21
(x+3)=2(y+3)
We solve this system of equations by susbstitution method.
x=21-y
[(21-y)+3]=2(y+3)
24-y=2y+6
-y-2y=6-24
-3y=-18
y=-18/-3=6
x=21-y
x=21-6=15.
Answer: Tom is 15 years old, and Tim is 6 years old.
Hope this helps you! =)
Sales at a local ice cream shop went 80% in 5 years. If 12,000 ice cream cones were sold in the current year find the number of ice cream cones sound in 5 years
Answers
"Sales at a local ice cream shop went 80% in 5 years. If 12,000 ice cream cones were sold in the current year find the number of ice cream cones sound in 5 years" should be "Sales at a local ice cream shop increased by 80% over 5 years. If 12,000 ice cream cones were sold in the current year find the number of ice cream cones sold in 5 years."
What y ou need to do here is to multiply "12,000 ice cream cones" by 1.80, which will produce a result including the original 12,000 cones PLUS 80% more cones:
1.80(12,000 cones) = 21,600 cones.
PLEASE HELP!!!!!! DUE TODAY!!!!!! PLEASEEEEEEEE!!!!!!!!
Ke Shawn is placing a right triangle in the coordinate plane. He knows that the longer leg is twice the length of the shorter leg. He places the longer leg on the x-axis.
What coordinates should he assign to the third vertex of the triangle?
Enter your answer in the boxes.
Answers
Answer: (2a,0) is the answer. But you already know.
Step-by-step explanation:
A lawn is in the shape of a trapezoid with a height of 40 feet and bases of 60 feet and 140 feet. How many bags of fertilizer must be purchased to cover the lawn if each bag covers 1800 square feet?
Answers
Rose bought 7/20 kilogram of ginger candy and 0.4 kilogram of cinnamon candy. Which did she buy more of? Explain how you know.
Answers
Answer:
Rose bought cinnamon candy more of.
Step-by-step explanation:
Rose bought ginger candy = [tex]\frac{7}{20}[/tex] kilogram
= 0.35 Kilogram
She bought cinnamon candy = 0.4 kilogram
As we know 1 kilogram = 1000 grams
0.35 kilogram = 1000 × 0.35 = 350 grams ginger candy
0.4 kilograms = 1000 × 0.4 = 400 grams cinnamon candy
Therefore, She bought cinnamon candy more.
6x+24/x^2+7x+12 simplify
Answers
6x+24/x^2+7x+12
becomes
6(x+4) 6(x+4)
------------------ = ---------------- Cancel the factors (x+4)
(x^2+7x+12) (x+4)(x+3)
6
= -------------
x+3
7,209,000 in the year 2050 and that this will be 24% of the population. What is the total population?
Answers
7,209,000/24% =
= 7,209,000/0.24
= 30,037,500
PLZ HELP IM FAILING!!!!!! The linear equation y−3=4(x+5)y−3=4(x+5) y minus 3 equals . 4 open x plus 5 close is in what form.
Answers
y-3 = 4(x+5) is "the equation of a straight line in point-slope form." That's it.
Only the Fahrenheit one plz
Answers
F = 1.8C + 32
The temperature C is given as -20
So, all you have to do is plug the given C temperature in the formula to get the corresponding F temperature as follows:
F = 1.8(-20) + 32 = -36 + 32 = -4 degrees Fahrenheit
Find f`(x) for the following function. Then find f(5),f(0),f(-9).
f(x)=2x+1 (simplify)
Answers
f(5)=11, f(0)=1 and f(-9)=-17.
We can calculate the gradient using these values:
Gradient=(11-1)/(5-0)=(-17-1)/(-9-0)=(-17-11)/(-9-5)=2.
What is the conversion factor to convert cups to liters?
Answers
Answer:
0.2365882365 L/cup
Step-by-step explanation:
The conversion factor can be found from the size of a cup in relation to a gallon, and the size of a cubic inch in relation to a liter.
[tex]1\,\text{cup}=8\,\text{fl oz}\times\dfrac{231\,in^{3}}{128\,\text{fl oz}}\times\left(\dfrac{0.254\,\text{dm}}{1\,\text{in}}\right)^{3}\times\dfrac{1\,L}{1\,dm^{3}}\\\\=\dfrac{8\cdot 231\cdot 0.254^{3}}{128}\,L=0.2365882365\,L[/tex]
By decreasing each dimension by 1 unit, the area of a rectangle decreased from 40 square feet (on the left) to 28 square feet (on the right). Find the percent decrease in area.
Answers
The percent decrease in the area of a rectangle from 40 square feet to 28 square feet is calculated by dividing the difference in areas (12 sq ft) by the original area (40 sq ft) and multiplying by 100%. The result is a 30% decrease in area.
Explanation:The percent decrease in the area of a rectangle can be found by first calculating the difference in the areas and then dividing that difference by the original area. In this case, the original area of the rectangle was 40 square feet and it decreased to 28 square feet, a difference of 12 square feet.
We use the formula for percentage change which is:
(change in value) / (original value) × 100%
Here, the change in value is the difference in areas (12 square feet) and the original value is the original area of the rectangle (40 square feet). Thus we calculate:
(12 sq ft/ 40 sq ft) * 100% = 30%
So, the percent decrease in the area of the rectangle is 30%.
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Jana is doing an experiment. She is on a dock that is 10 feet above the surface of the water. Jana drops the weighed end of a fishing line 35 feet below the surface of the water. She reels out the line 29 feet, and then reels it back in 7 feet. What is the final distance between Jana and the end of the finishing line?
Answers
Assumption: Line of reference is the surface of the water.
Final distance between Jana & fishing line = (dock height) +(reels out) -(reels back)
Final distance between Jana & fishing line = 10 + 35+ 29 - 7 = 67 feet
ANSWER: 67 feet
A data set a has a mean of 7,a median 5,and a mode of 8.wich of the the numbers 7,5,and 8 must be in the data set?
Answers
The height of Niagara falls is 182.How many yards high is Niagara falls
Answers
To convert the height of Niagara Falls from meters to yards, we multiply the given height of 182 meters by the conversion factor of 1.09361 yards per meter, which gives us approximately 199.18 yards.
Explanation:The height of Niagara Falls is provided in this question as 182. However, no units are given so let's assume this is in meters as that's a common unit for height. The conversion factor from meters to yards is approximately 1 meter = 1.09361 yards. Therefore, we multiply the given height in meters by this conversion factor to get the height in yards.
First, identify the height given in the question- here it is 182.Next, apply the conversion factor: 182 meters * 1.09361 yards/meter.Doing this multiplication gives us approximately 199.18 yards.So, Niagara Falls is approximately 199.18 yards high when converted from meters.
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The length of a ribbon is 3/4 meter. Sun Yi needs pieces measuring 1/3 meter for an art project. What is the greatest number of pieces measuring 1/3 meter that can be cut from the ribbon? How much ribbon will be left after Sun Yi cuts the ribbon?
Answers
Which of the following is true?
1.75 > 1½
0.25 ≠ ¼
0.2185 = ⅓
1.436 < 25⁄2
Answers
After converting all values to decimal form for consistent comparison, the statements '1.75 > 1½' and '1.436 < 25⁄2' are true, while the statements '0.25 ≠ ¼' and '0.2185 = ⅓' are false.
Explanation:The question is asking to compare different given values, either greater than (>), not equal to (≠), or less than (<). To do this accurately, we need to convert all numbers to a common form. In this case, it is easiest to consider everything as decimal values:
1.75 is indeed greater than 1½ (which is 1.5 in decimal), so 1.75 > 1½ is true.0.25 is equal to ¼ (which is also 0.25 in decimal), therefore 0.25 ≠ ¼ is false. 0.2185 is not equal to ⅓. ⅓ in decimal is approximately 0.333, so 0.2185 = ⅓ is false. 1.436 is less than 25/2 (which is 12.5 in decimal form), so 1.436 < 25⁄2 is true. Learn more about Decimal Comparison here:https://brainly.com/question/12970629
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The Clemson Tiger Football team won the 2016 College Football National Championship. During the regular season, they won 11 games and lost 1. The data set below is the number of points by which they won (positive values) or lost (negative value) each game.
{6,6,59,19,6,46,7,3,54,−1,22,49} = {-1,3,6,6,6,7,19,22,46,49,54,59}
What is the median? ________
What is the mode? _________
What is the mean? _________
What is the 5% trimmed mean? (Round to the nearest whole number when determining the number of values to trim).
Are these averages parameters or statistics?
Answers
median is the middle value, since there is an even number, add the 2 middle numbers and divide by 2
median = 9 +17 = 26/2 = 13
mode is the number that is repeated most often
mode = 6
mean is the average of all the numbers, when added they equal 276
276/12 = 23
mean = 23
5% trimmed mean = 218 /10 = 21.8
the averages are statistics since the points of a football game can vary
How do I Factor 125x^3 -64
The answer is not (5x+4)(5x-4) or (5x-4)(25x^2 +15x+16)
Answers
The correct factorization would be (5x - 4)(25x^2 + 20x + 16).
The law of factoring cubes states that we use the following formula when we have two cubes being subtracted by one another.
(a - b)(a^2 + ab + b^2)
If we take the cube root of 125x^3 we get 5x, which will be our a value. If we take the cube root of 64, we get 4, which will be our b value. Then we plug in to the formula.
(5x - 4)((5x)^2 + (5x)(4) + 4^2)
(5x - 4)(25x^2 + 20x + 16)
I think the problem with your second answer is that you just did not properly multiply the 5x and the 4 to get 20x (instead you got 15x).
How do i solve this equation and how did you do it
Answers
then f(1) = (3(1) + 7)²
= 10²
⇒ f(1) = 100 [OPTION E]
what is the mean of this data if necessary round your answer to the nearest tenth. please help with my homework
Answers
12 + 16 + 22 + 23 + 34 + 44 + 46 + 47 + 48 + 64 + 67 + 73 + 83 + 88 + 89 = 756
Then divide the answer by the total number of values
Number of values = 15
756 / 15 = 50.4
A suit is on sale for $154. If the original price of the suit was $440, what percent was discounted for the sale?
Answers
so, if we take 440 to be the 100%, what is 286 off of it in percentage anyway?
[tex]\bf \begin{array}{ccll} amount&\%\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 440&100\\ 286&p \end{array}\implies \cfrac{440}{286}=\cfrac{100}{p}\implies p=\cfrac{286\cdot 100}{440}[/tex]
How can you check to see if this equation is correct? 12+39=49
Answers
Answer:
By solving it.
Step-by-step explanation:
In order to see if the equation is correct, you just have to perform the operations in both sides of the inequality and then check if those are equal, that means that the equation is correct, for example if both sides of the equation give as a result 10 then the equation is correct, now let´s do it with the problem:
12+39=49
51=39
Since 51 is not equal to 39 that means that the equation is not correct.
A 16 oz package of brown rice cost 79cents and 32oz package of white rice costs $3.49 which package is a better deal
Answers
0.79 / 16 oz = 0.049 ( price for 1 oz of brown rice)
3.49 / 32 = 0.109 ( price for 1 oz of white rice)
the price per oz of brown rice is cheaper then 1 oz of white rice so the brown rice is the better deal
The 16 oz package of brown rice is a better deal because it has a lower price per ounce compared to the 32 oz package of white rice.
Explanation:To determine which package of rice is a better deal, we calculate the price per ounce for each package. The 16 oz package of brown rice costs 79 cents, so the price per ounce of brown rice is 79 cents ÷ 16 oz = 4.9375 cents per ounce. The 32 oz package of white rice costs $3.49, therefore the price per ounce of white rice is $3.49 ÷ 32 oz = 10.90625 cents per ounce.
By comparing the price per ounce, we can see that the price per ounce for the brown rice is less than the price per ounce for the white rice. Therefore, the 16 oz package of brown rice is the better deal.
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How much money should be invested at 5% interest, compounded quarterly, so that 7 years later the investment will be worth $5000? This amount is called the present value of $5000 at 5% interest.
Answers
identify the product and it in scientific notation. 4.7 x(6.02 x 10^23)
Answers
= (4.7 x 6.02)(4.7 x 10^23)
= (2.8294 x 10^1)(4.7 x 10^23)
= (2.8294 x 4.7)(10^1 x 10^23)
= 13.29818 x 10^24
= 1.329818 x 10^25
the product of [tex]\( 4.7 \times (6.02 \times 10^{23}) \) is \( 2.8294 \times 10^{24} \).[/tex]
Sure, let's identify the product and express it in scientific notation.
[tex]\( 4.7 \times (6.02 \times 10^{23}) \)[/tex]
To find the product, we simply multiply the two numbers:
[tex]\( 4.7 \times 6.02 \times 10^{23} \)[/tex]
First, we multiply 4.7 by 6.02:
[tex]\( 4.7 \times 6.02 = 28.294 \)[/tex]
Next, we multiply this result by [tex]\( 10^{23} \):[/tex]
[tex]\( 28.294 \times 10^{23} \)[/tex]
Now, to express this product in scientific notation, we need to adjust the decimal point so that the number is between 1 and 10:
[tex]\( 2.8294 \times 10^{24} \)[/tex]
Therefore, the product of [tex]\( 4.7 \times (6.02 \times 10^{23}) \) is \( 2.8294 \times 10^{24} \).[/tex]
1. First, we multiplied 4.7 by 6.02 to get 28.294.
2. Then, we multiplied this result by [tex]\( 10^{23} \)[/tex] to shift the decimal point 23 places to the right, resulting in[tex]\( 2.8294 \times 10^{24} \)[/tex] in scientific notation.
This method helps in handling large numbers efficiently, especially when dealing with calculations involving scientific notation and significant figures.
complete question
identify the product and it in scientific notation. 4.7 x(6.02 x 10^23)
For the initial value problem y' −xy = x, y(0) = 1. write down the first four nonzero terms of the series
Answers
The exact solution is easy enough to find. We can separate the variables:
[tex]y'-xy=x\iff \dfrac{\mathrm dy}{\mathrm dx}=x(1+y)\implies \dfrac{\mathrm dy}{1+y}=x\,\mathrm dx[/tex]
[tex]\implies\ln(1+y)=\dfrac{x^2}2+C[/tex]
[tex]\implies y=Ce^{x^2/2}-1[/tex]
Given that [tex]y(0)=1[/tex], we have
[tex]1=C-1\implies C=2[/tex]
Then recall that
[tex]e^x=\displaystyle\sum_{n\ge0}\frac{x^n}{n!}[/tex]
to write the solution as
[tex]y=2\displaystyle\sum_{n\ge0}\frac{\left(\frac{x^2}2\right)^n}{n!}-1[/tex]
[tex]y=1+2\displaystyle\sum_{n\ge1}\frac{\left(\frac{x^2}2\right)^n}{n!}[/tex]
so the first four terms of the series are
[tex]1+x^2+\dfrac{x^4}4+\dfrac{x^6}{24}[/tex]
6.24 divided by 200 equals what
Answers
A school uses merit based pay for teachers. Each teacher would be given a score between 1 (Far Below Expectations) and 5 (Exceeds expectations) in the following categories. The weight of each category is included.
Content Knowledge (20%)
Lesson plans (15%)
Instruction delivery (20%)
Classroom management (15%)
Student Performance (20%)
Supplementary Responsibilities (10%)
A teacher received the following scores. What was his weighted average?
Content Knowledge = 4
Lesson plans = 3
Instruction delivery = 3
Classroom management = 5
Student Performance = 4
Supplementary Responsibilities = 2
The teacher's weighted mean is ______________.
Answers
4*0.20 = 0.8
3 *0.15 = 0.45
3*0.20 = 0.6
5 *0.15 = 0.75
4 *.20 = 0.8
2 *0.10 =0.2
0.8 + 0.45 + 0.6 +0.75 +0.8 +0.2 = 3.6
weighted mean = 3.6
The teacher's weighted mean is 3.6 out of 5.
To calculate the weighted average, you'll multiply each score by its corresponding weight, then sum up the results.
Here are the scores and their respective weights:
- Content Knowledge: 4 (score) * 0.2 (weight) = 0.8
- Lesson Plans: 3 * 0.15 = 0.45
- Instruction Delivery: 3 * 0.20 = 0.6
- Classroom Management: 5 * 0.15 = 0.75
- Student Performance: 4 * 0.20 = 0.8
- Supplementary Responsibilities: 2 * 0.10 = 0.2
Now, sum up these weighted scores:
0.8 + 0.45 + 0.6 + 0.75 + 0.8 + 0.2 = 3.6
So, the teacher's weighted mean is 3.6.