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Cable theory is a mathematical model used to calculate currents and voltages along and across axon membranes. The model treats small sections of the membrane as the resistor-capacitor circuit shown in Figure 1. An electric neuronal signal enters the circuit through the input node. A portion of the signal remains inside the axon and continues to the next section of membrane via Output 1, and the rest of the signal exits to the extracellular fluid via Output 2.
Figure 1 Circuit unit used to model axon membranesResearchers studied cable theory by recreating the circuit in Figure 1 using a 2 μF capacitor for C_M, a 200 Ω resistor for R_M, and a 1,000 Ω resistor for R_L. The researchers sent an electrical signal through the input and discovered that the capacitor acted as a resistor whose resistance varied with the signal's frequency, as shown in Figure 2.
Figure 2 Capacitor equivalent resistance vs. signal frequency for C_M
Adapted from Tuckwell H, Introduction to Theoretical Neurobiology 2006 SIAM.
-To test the integrity of the capacitor, researchers maintained a voltage of 50 mV across the parallel portion of the circuit for several minutes. During this time, how much energy was stored in the capacitor?

Question

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Cable theory is a mathematical model used to calculate currents and voltages along and across axon membranes. The model treats small sections of the membrane as the resistor-capacitor circuit shown in Figure 1. An electric neuronal signal enters the circuit through the input node. A portion of the signal remains inside the axon and continues to the next section of membrane via Output 1, and the rest of the signal exits to the extracellular fluid via Output 2.

Figure 1 Circuit unit used to model axon membranesResearchers studied cable theory by recreating the circuit in Figure 1 using a 2 μF capacitor for C_M, a 200 Ω resistor for R_M, and a 1,000 Ω resistor for R_L. The researchers sent an electrical signal through the input and discovered that the capacitor acted as a resistor whose resistance varied with the signal's frequency, as shown in Figure 2.

Figure 2 Capacitor equivalent resistance vs. signal frequency for C_M
Adapted from Tuckwell H, Introduction to Theoretical Neurobiology 2006 SIAM.
-To test the integrity of the capacitor, researchers maintained a voltage of 50 mV across the parallel portion of the circuit for several minutes. During this time, how much energy was stored in the capacitor?

Quizplus · Accepted Answer

11ecba2d_12f2_5a06_a3bf_6557860a3e16_MD0008_00 Capacitance is the ability of a capacitor to store electric charges and is measured in farads (F). The simplest capacitor is a parallel plate capacitor, which consists of two evenly spaced, flat metallic plates. When a voltage is applied across a parallel plate capacitor, the plate connected to the higher voltage accumulates positive charges, and the plate connected to the lower voltage accumulates an equal amount of negative charges. The energy U in a capacitor is stored in the electric field created between the oppositely charged plates. Its magnitude depends on the capacitance C and the voltage V across the plates of the capacitor: 11ecba2d_12f2_5a07_a3bf_799187478b6d_MD0008_00 The capacitance is given as 2 μF, and the voltage is given as 50 mV. Using 1 μF = 10⁻⁶ F and 1 mV = 10⁻³ V: 11ecba2d_12f2_5a08_a3bf_3d5a5ea3bf01_MD0008_00 11ecba2d_12f2_5a09_a3bf_f5fe85ac6649_MD0008_00 V Using the above values in the capacitor energy equation: 11ecba2d_12f2_5a0a_a3bf_5b592cc45b68_MD0008_00 (see detailed calculation steps). The energy stored in the capacitor is 2.5 × 10⁻⁹ J. (Choice D) The amount of charge stored in a capacitor is Q = CV. If this equation is incorrectly used to calculate the charge instead of the energy stored in the capacitor, the result is 1.0 × 10⁻⁷ C. Educational objective: Capacitance is the ability of a capacitor to store electric charges. The amount of energy U stored in a charged capacitor depends on its capacitance C and the voltage V across its plates: 11ecba2d_12f2_5a0b_a3bf_516a45e117f6_MD0008_00 __________

Passage Cable Theory Is a Mathematical Model Used to Calculate Currents