3. A uniform magnetic field pointing out of the page has a changing field strength B(t) = at² + c in teslas and t time in seconds where a and c are constants. This magnetic field has a circular cross-section of radius R. (a) State the expression for Faraday's law. Write down its connection in terms of electric and magnetic fields starting with electromotive force, E. (b) Determine the magnitude of electric field at t = 2 seconds and position from center of cross section at r = (1/2)R.

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### Magnetic Fields and Induced Electric Fields

#### Problem Description
A uniform magnetic field pointing out of the page has a changing field strength given by the equation:
\[ B(t) = at^2 + c \] 
in teslas, where \( t \) is the time in seconds, and \( a \) and \( c \) are constants. This magnetic field has a circular cross-section of radius \( R \).

#### Questions

**(a)** State the expression for Faraday's law. Write down its connection in terms of electric and magnetic fields starting with the electromotive force, \( \mathcal{E} \).

**(b)** Determine the magnitude of the electric field at \( t = 2 \) seconds and the position from the center of the cross-section at \( r = \frac{1}{2}R \).

#### Solutions

**(a) Faraday’s Law**

Faraday's law of induction states that the electromotive force (EMF) around a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it is expressed as:
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]
where \(\Phi_B\) is the magnetic flux, defined as:
\[ \Phi_B = \int_{S} \mathbf{B} \cdot d\mathbf{A} \]
For a uniform magnetic field with area \( A \), the flux is:
\[ \Phi_B = B \cdot A \]

In terms of the relationship between electric fields and magnetic fields, the induced EMF can be expressed as:
\[ \mathcal{E} = \oint \mathbf{E} \cdot d\mathbf{l} \]
where \(\mathbf{E}\) is the induced electric field and \( d\mathbf{l} \) is the differential length element around the loop.

**(b) Electric Field Magnitude at Specific Time and Position**

Given:
\[ B(t) = at^2 + c \]

First, calculate the rate of change of the magnetic field:
\[ \frac{dB(t)}{dt} = \frac{d}{dt}(at^2 + c) = 2at \]

At \( t = 2 \) seconds:
\[ \frac{dB(t)}{dt} \big|_{t=2} =
Transcribed Image Text:### Magnetic Fields and Induced Electric Fields #### Problem Description A uniform magnetic field pointing out of the page has a changing field strength given by the equation: \[ B(t) = at^2 + c \] in teslas, where \( t \) is the time in seconds, and \( a \) and \( c \) are constants. This magnetic field has a circular cross-section of radius \( R \). #### Questions **(a)** State the expression for Faraday's law. Write down its connection in terms of electric and magnetic fields starting with the electromotive force, \( \mathcal{E} \). **(b)** Determine the magnitude of the electric field at \( t = 2 \) seconds and the position from the center of the cross-section at \( r = \frac{1}{2}R \). #### Solutions **(a) Faraday’s Law** Faraday's law of induction states that the electromotive force (EMF) around a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it is expressed as: \[ \mathcal{E} = -\frac{d\Phi_B}{dt} \] where \(\Phi_B\) is the magnetic flux, defined as: \[ \Phi_B = \int_{S} \mathbf{B} \cdot d\mathbf{A} \] For a uniform magnetic field with area \( A \), the flux is: \[ \Phi_B = B \cdot A \] In terms of the relationship between electric fields and magnetic fields, the induced EMF can be expressed as: \[ \mathcal{E} = \oint \mathbf{E} \cdot d\mathbf{l} \] where \(\mathbf{E}\) is the induced electric field and \( d\mathbf{l} \) is the differential length element around the loop. **(b) Electric Field Magnitude at Specific Time and Position** Given: \[ B(t) = at^2 + c \] First, calculate the rate of change of the magnetic field: \[ \frac{dB(t)}{dt} = \frac{d}{dt}(at^2 + c) = 2at \] At \( t = 2 \) seconds: \[ \frac{dB(t)}{dt} \big|_{t=2} =
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