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This rectangular loop is rotating at 6000 rpm in a uniform magnetic flux density of B = y 50 mT. Determine the current induced in the loop if its internal resistance is 0.5 92.

This rectangular loop is rotating at 6000 rpm in a uniform magnetic flux density of B = y 50 mT. Determine the current induced in the loop if its internal resistance is 0.5 92.

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### Problem Description This rectangular loop is rotating at 6000 rpm in a uniform magnetic flux density of \( B = y \, 50 \, \text{mT} \). Determine the current induced in the loop if its internal resistance is \( 0.5 \, \Omega \). ### Diagram Explanation - **Axes**: The diagram is drawn in a 3D coordinate system with labeled axes \( x \), \( y \), and \( z \). - **Loop**: There is a rectangular loop oriented in the \( yz \)-plane with its center at the origin. - **Dimensions**: The sides of the rectangle are both \( 3 \, \text{cm} \). - **Magnetic Field**: There is a uniform magnetic field, denoted by \( \mathbf{B} \), shown as arrows pointing in the positive \( y \)-direction. - **Rotation**: The loop is rotating about the \( x \)-axis with angular velocity \( \omega \). - **Angle**: The angle \( \phi(t) \) illustrates the angular position of the loop as it rotates. ### Solution Steps 1. **Induced EMF Calculation**: Use Faraday’s Law of Electromagnetic Induction to calculate the induced electromotive force (EMF) in the loop. 2. **Current Calculation**: Use Ohm’s Law to determine the current induced in the loop by dividing the induced EMF by the internal resistance \( 0.5 \, \Omega \). ### Key Concepts - **Faraday’s Law**: The induced EMF in a loop is proportional to the rate of change of magnetic flux through the loop. - **Ohm’s Law**: Current through a resistor is equal to the voltage across it divided by its resistance. This setup is a classic problem in electromagnetism, demonstrating the principles of electromagnetic induction in a rotating loop within a magnetic field.
Transcribed Image Text:### Problem Description This rectangular loop is rotating at 6000 rpm in a uniform magnetic flux density of \( B = y \, 50 \, \text{mT} \). Determine the current induced in the loop if its internal resistance is \( 0.5 \, \Omega \). ### Diagram Explanation - **Axes**: The diagram is drawn in a 3D coordinate system with labeled axes \( x \), \( y \), and \( z \). - **Loop**: There is a rectangular loop oriented in the \( yz \)-plane with its center at the origin. - **Dimensions**: The sides of the rectangle are both \( 3 \, \text{cm} \). - **Magnetic Field**: There is a uniform magnetic field, denoted by \( \mathbf{B} \), shown as arrows pointing in the positive \( y \)-direction. - **Rotation**: The loop is rotating about the \( x \)-axis with angular velocity \( \omega \). - **Angle**: The angle \( \phi(t) \) illustrates the angular position of the loop as it rotates. ### Solution Steps 1. **Induced EMF Calculation**: Use Faraday’s Law of Electromagnetic Induction to calculate the induced electromotive force (EMF) in the loop. 2. **Current Calculation**: Use Ohm’s Law to determine the current induced in the loop by dividing the induced EMF by the internal resistance \( 0.5 \, \Omega \). ### Key Concepts - **Faraday’s Law**: The induced EMF in a loop is proportional to the rate of change of magnetic flux through the loop. - **Ohm’s Law**: Current through a resistor is equal to the voltage across it divided by its resistance. This setup is a classic problem in electromagnetism, demonstrating the principles of electromagnetic induction in a rotating loop within a magnetic field.
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Step 1: Determine the given data:

Advanced Physics homework question answer, step 1, image 1

straight omega equals 6000 space rpm equals 6000 cross times fraction numerator 2 straight pi over denominator 60 end fraction equals 200 straight pi space rad divided by straight s straight B equals 50 cross times 10 to the power of negative 3 end exponent bold y space straight T straight R equals 0.5 space straight capital omega straight A equals left parenthesis 2 cross times 3 right parenthesis equals 6 space cm squared equals 6 cross times 10 to the power of negative 4 end exponent space straight m squared

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