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g) Show the time rate of change in the magnetic flux is d$m(x) _ _Ho'l •a ·b• V [ dt [9 + x) • x. h) Use Faradays Law to determine the induced EMF at x = 3cm

g) Show the time rate of change in the magnetic flux is d$m(x) _ _Ho'l •a ·b• V [ dt [9 + x) • x. h) Use Faradays Law to determine the induced EMF at x = 3cm

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Please answer G, H. Thank you

The following picture shows a LONG conductor carrying current I. Nearby there is a conducting rectangular loop with sides a = 8 cm and b = 4 cm. The loop also caries a resistance R = 10 ohms. The curent is constant and has a value of I = 6.0 Amperes. The loop is moving away to the right with a constant velocity, V = 2 m/s. Answer the following questions at the instant of time t' when the left edge of the loop is at position x" as shown below Use the coordinate system , x to the right, y into the board, z upward a) Write an expression for the magnetic field as a function of the distance "x" (from the LONG conductor to the loop. ) USE “+" for CCW circulation and "-“ for CW circulation. b) Write the magnetic field in "i-j-k" format at point "x"to the right of the current carrying wire in the "i-z" plane c) Write the infinitesimal area vector for the loop in "i-j-k" format d) Write the explicit integral for the magnetic flux through the area of the loop using the answer for B and dA above. Include limits on the area integral In general = || 5 - aa e) Integrate the flux integral above to obtain „(x) = 40""[In(x + b) – In(x)] f) Express the position "x" as a function of r g) Show the time rate of change in the magnetic flux is d®m(x) _ _Ho•I •a·b•V[ dt x-(x+ b)] h) Use Faradays Law to determine the induced EMF at x = 3cm
Transcribed Image Text:The following picture shows a LONG conductor carrying current I. Nearby there is a conducting rectangular loop with sides a = 8 cm and b = 4 cm. The loop also caries a resistance R = 10 ohms. The curent is constant and has a value of I = 6.0 Amperes. The loop is moving away to the right with a constant velocity, V = 2 m/s. Answer the following questions at the instant of time t' when the left edge of the loop is at position x" as shown below Use the coordinate system , x to the right, y into the board, z upward a) Write an expression for the magnetic field as a function of the distance "x" (from the LONG conductor to the loop. ) USE “+" for CCW circulation and "-“ for CW circulation. b) Write the magnetic field in "i-j-k" format at point "x"to the right of the current carrying wire in the "i-z" plane c) Write the infinitesimal area vector for the loop in "i-j-k" format d) Write the explicit integral for the magnetic flux through the area of the loop using the answer for B and dA above. Include limits on the area integral In general = || 5 - aa e) Integrate the flux integral above to obtain „(x) = 40""[In(x + b) – In(x)] f) Express the position "x" as a function of r g) Show the time rate of change in the magnetic flux is d®m(x) _ _Ho•I •a·b•V[ dt x-(x+ b)] h) Use Faradays Law to determine the induced EMF at x = 3cm
Expert Solution
Step 1

Faraday's Law of Electromagnetic Induction

  • Whenever there is a change in magnetic flux linked to a coil, an emf is induced in the coil.
  • The induced emf is equal to the rate of change of magnetic flux.

 

Lenz's Law

The direction of the induced emf is such that it opposes the very cause for which it has been formed.

If dϕ is the change in magnetic flux in time dt then the induced emf is

ε=-dϕdt

Step 2

g) In question (e) you have found that the magnetic flux linked to the rectangular loop is

ϕm=μ0Ia2πlnx+b-ln(x)

Taking the time derivative of the magnetic flux

dϕmdt=μ0Ia2πddtlnx+b-lnx=μ0Ia2π1x+bdxdt-1xdxdt=μ0Ia2πdxdt1x+b-1x=μ0Iav2πx-x-bxx+b=-μ0Iabv2π1x(x+b)

dxdt=v is the velocity of the rectangular loop.

 

h) Now from Faraday's Law we know that the induced emf

ε=-dϕmdt=μ0Iabv2π1xx+b

Using the data given in the question, at x=3 cm, the induced emf is

ε=4π×10-7×6×8×10-2×4×10-2×22π13×10-2×3×10-2+4×10-2=3.66×10-6 V=3.66 μV

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