Question

For what emf $\epsilon$does the $200\Omega$resistor in$figureP28.64$ dissipate no power? Should the emf be oriented with its positive terminal at the top or at the bottom?

Short answer

$\epsilon =150V$and the positive terminal is bottom

Step-by-step solution

Given Information

We should find oriented with its positive terminal at the top or at the bottom.

Simplify

When the current flows through a resistor, it dissipates energy, the rater of the dissipated energy is the power. This rate where the energy is transferred to the current to the resistor is

$P_R=I∆V_R$

The dissipated power is zero when the current through the resistor $200\Omega$is zero (no current flow). But at the condition, the current will not flow through the $200\Omega$? This occurs when the right battery $\epsilon$applies a current through $300\Omega$with the same magnitude of the left battery but in opposite direction. So, let us find the current that is flow due the left battery $50V$using Ohm's law.

Ohm's law states that the current Iflows through a resistance due to the potential difference V between resistance.

$I=\frac{V}{R}$

So, we get the current through $100\Omega$by using the values $50Vand100\Omega$into equation $\left(2\right)$

$I_{100\Omega }=\frac{V}{R}=\frac{50V}{100\Omega }=0.5A$

As we mentioned above, this is the current that flows in $300\Omega$to make no current flows through $200\Omega$. So, the voltage that exert this current through the resistor $300\Omega$is calculated by Ohm's law in the form

$$\begin{gathered} \epsilon =IR \\ =(0.5A)\left(300\Omega \right) \\ =150V \end{gathered}$$

The right battery has emf $\epsilon =150V$and to exerts a current in the opposite direction, its terminal positive should be bottom.