The circuit shown in Figure P 28.73 is set up in the laboratory to measure an unknown capacitanc

The circuit shown in Figure P 28.73 is set up in the...

The circuit shown in Figure $\mathrm{P} 28.73$ is set up in the laboratory to measure an unknown capacitance $C$ with the use of a voltmeter of resistance $R=10.0 \mathrm{M} \Omega$ and a battery whose emf is $6.19 \mathrm{V} .$ The data given in the table are the measured voltages across the capacitor as a function of time, where $t=0$ represents the instant at which the switch is opened. (a) Construct a graph of $\ln (\mathcal{E} / \Delta V)$ versus $t,$ and perform a linear least-squares fit to the data. (b) From the slope of your graph, obtain a value for the time constant of the circuit and a value for the capacitance. FIGURE CANT COPY

Key Concepts

Linear Least-Squares Fit

The linear least-squares fit is a statistical method used to determine the best-fitting straight line through a set of data points by minimizing the sum of the squares of the vertical deviations (residuals) from the line. This technique is essential for accurately estimating parameters from experimental data, such as the slope, which in the context of RC circuits can be directly related to the time constant of the circuit. It provides an objective means of handling experimental error and variations in the measurements.

Time Constant

The time constant, denoted by ?, is a key parameter in RC circuits defined by the product of the resistance and the capacitance (? = RC). It represents the characteristic time over which the capacitor charges or discharges; specifically, after one time constant, the voltage across a discharging capacitor will have dropped to about 37% of its initial value. This concept is vital for designing circuits with desired timing characteristics and for understanding transient responses.

Graphical Linearization

Graphical linearization is a technique used to transform an exponential relationship into a linear form. By taking the natural logarithm of both sides of an exponential decay equation, the data can be plotted as a straight line which facilitates the analysis and extraction of key parameters such as the slope and y-intercept. This method is widely used in experimental science to simplify the interpretation of data that follows a non-linear trend.

RC Circuit Analysis

RC circuit analysis involves understanding the behavior of circuits containing resistors and capacitors. These circuits are fundamental in electronics because they introduce time-dependent behavior, particularly in how voltages and currents change over time during charging and discharging processes. Analyzing such circuits typically requires using differential equations that describe the exponential nature of the voltage or current changes, as well as concepts like impedance and phase shift when alternating currents are involved.

Exponential Decay in RC Circuits

When a capacitor discharges through a resistor, the voltage across the capacitor decreases exponentially over time. This behavior is described by an equation of the form V(t) = V0 exp(-t/?), where V0 is the initial voltage at t = 0 and ? is the time constant of the circuit. This concept is crucial because it allows one to predict and analyze the dynamic response of the circuit based on its resistance and capacitance.

Transcript

00:01 Okay, so we have a circuit here that represents a setup to measure a no capacitance, okay, that it's here in the center of the circuit.

00:09 And what we use is we use a voltmeter, okay, this thing in here, that has a resistance of 10 mega -on, okay, that is connected to a battery that gives 6 .19 volts, okay? and what we do, or what we can measure then, it's how the potential difference changes when the switch is open in function of time.

00:41 That is what we have here, okay? so we are giving this set of data, we have time, and we have how the potential changes or the measure in the volt meter, okay? and for the first part is actually the one i have here.

00:56 I'm asked to construct a graph of the natural logarithm of e divided by delta v.

01:05 Okay? so this e is actually the value of the potential difference in the battery.

01:10 So what you have to do is divide 6 .19 for each one of these values of v.

01:18 Okay? and then take the natural logarithm of that and you will have as a result this set of data.

01:26 So i already did this.

01:28 Okay, we are going to understand what this means like eventually.

01:34 But for now, what i do then in a software, this is up to your choice.

01:39 I use microsoft excel.

01:41 You can use whatever you want.

01:42 Okay, is asked to plot time against this.

01:47 Okay, and you are going to get as a result these points, the blue points, okay, that i'm showing here.

01:54 And then you have to ask the software to do a linear fit of this set of data, okay? and you will get this dotted line, and then you have to ask for it to give you the equation of the line.

02:13 You will get something like this.

02:15 Okay, so i'm going to rewrite this and b.

02:18 You will get that y is 0 .0118.

02:24 X, okay, plus 0 .082.

02:33 Okay, so this is the equation of the line.

02:36 But then what does this mean? okay, so we go to part b, we are asked to find the value of the time constant and the capacitance, okay? so for this, remember that delta v as a function of time in a capacitor equals b0...

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