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In these notes, we will examine a few examples of changing magnetic fluxes and associated induced voltages. First let's consider when $\Phi_B$ rises and falls linearly with the same magnitude of slope:

From t = 0 to t = 5, $\Phi_B$ has a positive, constant slope, so Faraday's law tells us that $V_{ind}$ will be constant and negative. From t = 5 to t = 10, $\Phi_B$ has a positive, constant slope, so Faraday's law tells us that $V_{ind}$ will be constant and positive. $$ \Phi_B(t)= \begin{cases} 2t & \text{if } 0<t<5\\ -2t & \text{if } 5<t<10 \end{cases} $$

$$ \frac{d \Phi_B}{dt}= \begin{cases} 2 & \text{if } 0<t<5\\ -2 & \text{if } 5<t<10 \end{cases} $$

$$ V_{ind}= \begin{cases} -2 & \text{if } 0<t<5\\ 2 & \text{if } 5<t<10 \end{cases} $$

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