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title: "A non-ideal battery with constant electromotive force \\(\\mathcal{E}\\) has an internal resistance \\(r(t)\\) that increases linearly with time due to temperature buildup according to \\(r(t) = r_0(1 + \\beta t)\\), where \\(r_0\\) and \\(\\beta\\) are positive constants. At time \\(t = 0\\), the battery is connected to an external load resistor of constant resistance \\(R\\). Which of the following expressions represents the total energy \\(E_R\\) dissipated in the load resistor \\(R\\) from \\(t = 0\\) to \\(t = T\\)?"
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url: "https://nerd-notes.com/ubq/118451/"
date_modified: "2026-08-04T08:10:13+00:00"
---

# A non-ideal battery with constant electromotive force \(\mathcal{E}\) has an internal resistance \(r(t)\) that increases linearly with time due to temperature buildup according to \(r(t) = r_0(1 + \beta t)\), where \(r_0\) and \(\beta\) are positive constants. At time \(t = 0\), the battery is connected to an external load resistor of constant resistance \(R\). Which of the following expressions represents the total energy \(E_R\) dissipated in the load resistor \(R\) from \(t = 0\) to \(t = T\)?

A non-ideal battery with constant electromotive force \(\mathcal{E}\) has an internal resistance \(r(t)\) that increases linearly with time due to temperature buildup according to \(r(t) = r_0(1 + \beta t)\), where \(r_0\) and \(\beta\) are positive constants. At time \(t = 0\), the battery is connected to an external load resistor of constant resistance \(R\). Which of the following expressions represents the total energy \(E_R\) dissipated in the load resistor \(R\) from \(t = 0\) to \(t = T\)?

![A single-loop circuit schematic. On the left vertical branch is a non-ideal battery enclosed in a dashed rectangular box, containing a constant DC voltage source labeled \(\mathcal{E}\) in series with a resistor labeled \(r(t)\). On the right vertical branch is a single load resistor labeled \(R\). Top and bottom horizontal wires connect the battery terminals to the load resistor.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-circuit-1-1785831012-kuuv6q.jpg)

- **A.** \(\dfrac{\mathcal{E}^2 R T}{\left(R + r_0 + \dfrac{1}{2}r_0 \beta T\right)^2}\)
- **B.** \(\dfrac{\mathcal{E}^2 R T}{(R + r_0)(R + r_0 + r_0 \beta T)}\)
- **C.** \(\dfrac{\mathcal{E}^2 R T}{(R + r_0)(R + r_0 \beta T)}\)
- **D.** \(\dfrac{\mathcal{E}^2}{r_0 \beta} \ln\left(1 + \dfrac{r_0 \beta T}{R + r_0}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118451/*