1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
myrzilka [38]
1 year ago
14

A closely wound, circular coil with a diameter of 4.30 cm has 470 turns and carries a current of 0.460 A .

Physics
2 answers:
Nikolay [14]1 year ago
8 0
Part A is B= 5.65×10-3
Nadusha1986 [10]1 year ago
5 0

Hi there!

a)
Let's use Biot-Savart's law to derive an expression for the magnetic field produced by ONE loop.

dB = \frac{\mu_0}{4\pi} \frac{id\vec{l} \times \hat{r}}{r^2}

dB = Differential Magnetic field element

μ₀ = Permeability of free space (4π × 10⁻⁷ Tm/A)

R = radius of loop (2.15 cm = 0.0215 m)

i = Current in loop (0.460 A)

For a circular coil, the radius vector and the differential length vector are ALWAYS perpendicular. So, for their cross-product, since sin(90) = 1, we can disregard it.

dB = \frac{\mu_0}{4\pi} \frac{id\vec{l}}{r^2}

Now, let's write the integral, replacing 'dl' with 'ds' for an arc length:
B = \int \frac{\mu_0}{4\pi} \frac{ids}{R^2}

Taking out constants from the integral:
B =\frac{\mu_0 i}{4\pi R^2}  \int ds

Since we are integrating around an entire circle, we are integrating from 0 to 2π.

B =\frac{\mu_0 i}{4\pi R^2}  \int\limits^{2\pi R}_0 \, ds

Evaluate:
B =\frac{\mu_0 i}{4\pi R^2}  (2\pi R- 0) = \frac{\mu_0 i}{2R}

Plugging in our givens to solve for the magnetic field strength of one loop:

B = \frac{(4\pi *10^{-7}) (0.460)}{2(0.0215)} = 1.3443 \mu T

Multiply by the number of loops to find the total magnetic field:
B_T = N B = 0.00631 = \boxed{6.318 mT}

b)

Now, we have an additional component of the magnetic field. Let's use Biot-Savart's Law again:
dB = \frac{\mu_0}{4\pi} \frac{id\vec{l} \times \hat{r}}{r^2}

In this case, we cannot disregard the cross-product. Using the angle between the differential length and radius vector 'θ' (in the diagram), we can represent the cross-product as cosθ. However, this would make integrating difficult. Using a right triangle, we can use the angle formed at the top 'φ', and represent this as sinφ.  

dB = \frac{\mu_0}{4\pi} \frac{id\vec{l} sin\theta}{r^2}

Using the diagram, if 'z' is the point's height from the center:

r = \sqrt{z^2 + R^2 }\\\\sin\phi = \frac{R}{\sqrt{z^2 + R^2}}

Substituting this into our expression:
dB = \frac{\mu_0}{4\pi} \frac{id\vec{l}}{(\sqrt{z^2 + R^2})^2} }(\frac{R}{\sqrt{z^2 + R^2}})\\\\dB = \frac{\mu_0}{4\pi} \frac{iRd\vec{l}}{(z^2 + R^2)^\frac{3}{2}} }

Now, the only thing that isn't constant is the differential length (replace with ds). We will integrate along the entire circle again:
B = \frac{\mu_0 iR}{4\pi (z^2 + R^2)^\frac{3}{2}}} \int\limits^{2\pi R}_0, ds

Evaluate:
B = \frac{\mu_0 iR}{4\pi (z^2 + R^2)^\frac{3}{2}}} (2\pi R)\\\\B = \frac{\mu_0 iR^2}{2 (z^2 + R^2)^\frac{3}{2}}}

Multiplying by the number of loops:
B_T= \frac{\mu_0 N iR^2}{2 (z^2 + R^2)^\frac{3}{2}}}

Plug in the given values:
B_T= \frac{(4\pi *10^{-7}) (470) (0.460)(0.0215)^2}{2 ((0.095)^2 + (0.0215)^2)^\frac{3}{2}}} \\\\ =  0.00006795 = \boxed{67.952 \mu T}

You might be interested in
How much force is needed to accelerate a 20 kg mass at a rate of 4 m/s to the second power?
dusya [7]

Answer:

So 55 Newtons are needed.

6 0
3 years ago
Read 2 more answers
Which tem decorbes the perceived frequency of a sound wave?
JulsSmile [24]

Pitch

Explanation:

The pitch of a sound wave is the perceived frequency of a sound wave.

The quality of sound that makes it discernible to the ear is the pitch.

  • Sound wave is a longitudinal wave that is transmitted by series of rarefaction and compression.
  • Pitch helps to distinguish between the different sound qualities.
  • Pitch is how high or low sound is perceived.

learn more:

Pitch brainly.com/question/9772227

Sound wave brainly.com/question/2845448

#learnwithBrainly

5 0
3 years ago
Read 2 more answers
What factors determine the amount of air resistance an object experiences?
geniusboy [140]
The pressure of the air at the way its blowing

3 0
2 years ago
Name the type of component that has a greater resistance as the current through it increases​
Gnesinka [82]

Answer:

filament bulb, filament lamp

Explanation:

4 0
3 years ago
10. A loud motorcycle passes by the front of Bill's house. What medium
garik1379 [7]

Answer: Air

Explanation:

3 0
2 years ago
Read 2 more answers
Other questions:
  • Atmospheric pressure is reported in a variety of units depending on local meteorological preferences. In many European countries
    9·1 answer
  • In which one or more of the following is the earth assumed to be the center of the universe?
    15·2 answers
  • Which of the following describes a major role of gravity in star formation?
    12·2 answers
  • A shank has a moment of inertia about the knee joint of 0.18 kg∙m^2. Assume that the quadriceps muscles (vasti and the rectus fe
    5·1 answer
  • How can the rate of a reaction be increased?
    12·2 answers
  • The planet earth orbits around the sun and also spins around its own axis. True or False
    7·1 answer
  • The two most prominent wavelengths in the light emitted by a hydrogen discharge lamp are 656 nm (red) and 486 nm (blue). Light f
    10·1 answer
  • Match the following items.
    6·1 answer
  • Answer (soon please)
    8·1 answer
  • What type of data can be used to make a model of food?
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!