Questions: Electrical systems are governed by Ohm's law, which states that V=I R, where V= voltage, I= current, and R= resistance. If the current in an electrical system is decreasing at a rate of 7 A/S, while the voltage remains constant at 10 V at what rate is the resistance increasing when the current is 28 A. A. 0.4 Ω/s B. 0.1 Ω/s C. 2.5 Ω/s D. 17.5 Ω/s

Electrical systems are governed by Ohm's law, which states that V=I R, where V= voltage, I= current, and R= resistance. If the current in an electrical system is decreasing at a rate of 7 A/S, while the voltage remains constant at 10 V at what rate is the resistance increasing when the current is 28 A.
A. 0.4 Ω/s
B. 0.1 Ω/s
C. 2.5 Ω/s
D. 17.5 Ω/s

Transcript text: 12. [10 pt] Electrical systems are governed by Ohm's law, which states that $V=I R$, where $V=$ voltage, $I=$ current, and $R=$ resistance. If the current in an electrical system is decreasing at a rate of $7 \frac{\mathrm{~A}}{\mathrm{~S}}$, while the voltage remains constant at 10 V at what rate is the resistance increasing when the current is 28 A . A. $0.4 \frac{\Omega}{\mathrm{~s}}$ B. $0.1 \frac{\Omega}{\mathrm{~s}}$ C. $2.5 \frac{\Omega}{\mathrm{~s}}$ D. $17.5 \frac{\Omega}{\mathrm{~s}}$

Solution

Solution Steps

Step 1: Identify the given information and the relationship

We are given:

Voltage $ V = 10 $ V (constant)
Current $ I = 28 $ A
Rate of change of current $ \frac{dI}{dt} = -7 \frac{\mathrm{A}}{\mathrm{s}} $

We need to find the rate of change of resistance $ \frac{dR}{dt} $.

Step 2: Use Ohm's Law to express resistance

Ohm's Law states: \[ V = I R \] Since $ V $ is constant, we can differentiate both sides with respect to time $ t $: \[ \frac{dV}{dt} = \frac{d}{dt}(I R) \]

Step 3: Differentiate and solve for $\frac{dR}{dt}$

Since $ V $ is constant, $ \frac{dV}{dt} = 0 $: \[ 0 = I \frac{dR}{dt} + R \frac{dI}{dt} \]

Rearrange to solve for $ \frac{dR}{dt} $: \[ I \frac{dR}{dt} = -R \frac{dI}{dt} \] \[ \frac{dR}{dt} = -\frac{R}{I} \frac{dI}{dt} \]

Step 4: Calculate resistance $ R $

Using Ohm's Law: \[ R = \frac{V}{I} = \frac{10}{28} \approx 0.3571 \, \Omega \]

Step 5: Substitute values and solve for $\frac{dR}{dt}$

Substitute $ R = 0.3571 \, \Omega $, $ I = 28 \, \mathrm{A} $, and $ \frac{dI}{dt} = -7 \, \frac{\mathrm{A}}{\mathrm{s}} $: \[ \frac{dR}{dt} = -\frac{0.3571}{28} \times (-7) \] \[ \frac{dR}{dt} = \frac{0.3571 \times 7}{28} \] \[ \frac{dR}{dt} = \frac{2.4997}{28} \] \[ \frac{dR}{dt} \approx 0.0893 \, \frac{\Omega}{\mathrm{s}} \]