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Step 1:I'll solve this problem step by step, carefully following the LaTeX formatting guidelines:
Step 2:: Understand the Problem Setup
- Total resistance: $$R = 8.00 \Omega
- Number of loops: N = 20
Step 3:: Calculate the Coil Area
A = a \times b = (0.0800 \mathrm{~m}) \times (0.120 \mathrm{~m}) = 9.60 \times 10^{-3} \mathrm{~m}^{2}
Step 4:: Magnetic Flux Calculation
The magnetic flux $$\Phi = NBA \cos(\theta)$$, where $$\theta = 0°$$ (perpendicular to field)
\Phi = 20 \times 3.00 \mathrm{~T} \times (9.60 \times 10^{- 3} \mathrm{~m}^{2}) = 0.576 \mathrm{~Wb}
Step 5:: Induced EMF Calculation
4) 75% exiting field: $$\frac{d\Phi}{dt} = -\frac{3}{4} \times NBA \frac{v}{L}
Situations:
Step 6:: Detailed EMF Calculations
\mathcal{E} = -20 \times 3.00 \mathrm{~T} \times (9.60 \times 10^{-3} \mathrm{~m}^{2}) \times \frac{0.0500 \mathrm{~m/s}}{0.120 \mathrm{~m}} = -0.240 \mathrm{~V}
- Velocity v = 0.0500 \mathrm{~m/s} Half in field:
Step 7:: Current Calculation
Using Ohm's law: $$I = \frac{\mathcal{E}}{R}
I = \frac{- 0.240 \mathrm{~V}}{8.00 \Omega} = - 0.0300 \mathrm{~A} Lenz's Law Analysis: - Negative EMF indicates current will create a magnetic field opposing the change - Current will flow counterclockwise to create a magnetic field pointing out of the page
Final Answer
(a) Induced EMF and current values: - Outside field: \mathcal{E} = 0 \mathrm{~V}, I = 0 \mathrm{~A} - Half in field: \mathcal{E} = - 0.240 \mathrm{~V}, I = - 0.0300 \mathrm{~A} (counterclockwise) - Entirely in field: \mathcal{E} = 0 \mathrm{~V}, I = 0 \mathrm{~A} - 75% exiting: \mathcal{E} = + 0.240 \mathrm{~V}, I = + 0.0300 \mathrm{~A} (clockwise) (b) The induced EMF is the same when half or 25% of the coil enters because the rate of change of magnetic flux is proportional to the velocity and the portion of the coil in the field.
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