Let V_B be the rate of decrease of the magnetic fields(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{dB}{dt}[/tex]

For the 3rd path:

[tex]\oint E\cdot ds = -\frac{d\phi _B}{dt} = -\frac{{d\phi _B}_1 + {d\phi _B}_2}{dt}[/tex]

[tex]\phi _B_{(t)} = A_{(t)}B_{(t)}[/tex]

The area is constant, it's only the magnetic field that's changing:

[tex]\phi _B_{(t)} = \pi R^2(B_0 - V_Bt)[/tex]

Since B1 and B2 are in opposite directions, give one of them a minus sign:

[tex]{\phi _B}_1 + {\phi _B}_2 = \pi R_1^2(B_0 - V_Bt) - \pi R_2^2(B_0 - V_Bt) = \pi (B_0 - V_Bt)(R_1^2 - R_2^2)[/tex]

Take the derivative of that:

[tex]\frac{{d\phi _B}_1 + {d\phi _B}_2}{dt} = -\pi V_B(R_1^2 - R_2^2)[/tex]

And therefore:

[tex]\oint E\cdot ds = \pi V_B(R_1^2 - R_2^2)[/tex]

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Rate of decrease of the magnetic fields

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