Math

QuestionIf a wire creates a 10 T10 \mathrm{~T} field at distance rr, what is the net field at rr with another wire at 2r2r?

Studdy Solution

STEP 1

Assumptions1. The magnetic field a distance rr away from a current-carrying wire is 10 10 \mathrm{~} . Another wire is placed a distance rr from the original wire with rr in the middle3. The second wire has a current twice as strong flowing in the opposite direction4. The magnetic field due to a current-carrying wire at a distance rr is given by Ampere's law B=μ0IπrB = \frac{\mu0 I}{ \pi r}, where BB is the magnetic field, μ0\mu0 is the permeability of free space, $$ is the current, and $r$ is the distance from the wire5. The net magnetic field at a point is the vector sum of the magnetic fields due to each wire at that point

STEP 2

First, we need to find the current in the first wire. We can do this by rearranging Ampere's law to solve for $$.
=B×2πrμ0 = \frac{B \times2 \pi r}{\mu0}

STEP 3

Now, plug in the given values for the magnetic field and distance to calculate the current in the first wire.
=10 ×2πrμ0 = \frac{10 \mathrm{~} \times2 \pi r}{\mu0}

STEP 4

Next, we find the current in the second wire. According to the problem, this is twice the current in the first wire.
2=2=2×10 ×2πrμ02 =2 =2 \times \frac{10 \mathrm{~} \times2 \pi r}{\mu0}

STEP 5

Now, we calculate the magnetic field at a distance rr from the second wire using Ampere's law.
B2=μ0I22πrB2 = \frac{\mu0 I2}{2 \pi r}

STEP 6

Substitute the value of 22 from4 into the equation from5.
B2=μ0×2×10 ×2πrμ02πrB2 = \frac{\mu0 \times2 \times \frac{10 \mathrm{~} \times2 \pi r}{\mu0}}{2 \pi r}

STEP 7

implify the equation to find B2B2.
B2=20 B2 =20 \mathrm{~}

STEP 8

The net magnetic field at a distance rr from the first wire is the vector sum of the magnetic fields due to the first and second wires. Since the currents are in opposite directions, the magnetic fields will subtract.
Bnet=BB2B_{net} = B - B2

STEP 9

Substitute the values of BB and B2B2 into the equation from8.
Bnet= 20 B_{net} = \mathrm{~} -20 \mathrm{~}

STEP 10

Calculate the net magnetic field.
Bnet=10 B_{net} = -10 \mathrm{~}Since magnetic field is a vector quantity, the negative sign indicates that the net magnetic field is in the opposite direction to the field due to the first wire. However, in terms of magnitude, the net magnetic field is 10 10 \mathrm{~}.
So, the answer is not listed among the options given in the problem. There might be a mistake in the problem or the options provided.

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