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Problem 3: It is postulated that the radial electric field of a group of charges falls off as E, = C/r", where C is a constant, r is the distance from the center of the group, and n is an unknown exponent. To test this hypothesis, you make a field probe costisting of two needle tips spaced Ar = 1.0 mm apart. You orient the needles so that a line between the tips points to the center of the charges, then use a voltmeter to read the potential difference between the tips. After you take measurements at several distances from the center of the group, your data are as given in the table. Use an appropriate graph of the data to determine the constants C and n. Distance, r (cm) Potential difference, AV (mV) 34.7 6.6 2.1 1.2 0.6 2.0 4.0 6.0 8.0 10.0

Problem 3: It is postulated that the radial electric field of a group of charges falls off as E, = C/r", where C is a constant, r is the distance from the center of the group, and n is an unknown exponent. To test this hypothesis, you make a field probe costisting of two needle tips spaced Ar = 1.0 mm apart. You orient the needles so that a line between the tips points to the center of the charges, then use a voltmeter to read the potential difference between the tips. After you take measurements at several distances from the center of the group, your data are as given in the table. Use an appropriate graph of the data to determine the constants C and n. Distance, r (cm) Potential difference, AV (mV) 34.7 6.6 2.1 1.2 0.6 2.0 4.0 6.0 8.0 10.0

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Hello I really need help with part A please that is all I need is help with part A

**Problem 3:** It is postulated that the radial electric field of a group of charges falls off as \( E_r = \frac{C}{r^n} \), where \( C \) is a constant, \( r \) is the distance from the center of the group, and \( n \) is an unknown exponent. To test this hypothesis, you make a field probe consisting of two needle tips spaced \(\Delta r = 1.0 \, \text{mm}\) apart. You orient the needles so that a line between the tips points to the center of the charges, then use a voltmeter to read the potential difference between the tips. After you take measurements at several distances from the center of the group, your data are as given in the table. Use an appropriate graph of the data to determine the constants \( C \) and \( n \). | Distance, \( r \) (cm) | Potential difference, \( \Delta V \) (mV) | |------------------------|------------------------------------------| | 2.0 | 34.7 | | 4.0 | 6.6 | | 6.0 | 2.1 | | 8.0 | 1.2 | | 10.0 | 0.6 | a) In this experiment you are measuring the change of potential \( \Delta V \) over small displacement \( \Delta r \). In the limit \( \Delta r \to 0 \), \( \Delta V/\Delta r \) would give you the derivative of \( V \) with respect to \( r \), which is the radial component of the electric field, \( E_r \) (strictly speaking, with the minus sign). Take the natural logarithm of both sides of \( \frac{\Delta V}{\Delta r} = \frac{C}{r^n} \) to show that \(\ln\left(\frac{\Delta V}{\Delta r}\right)\) can be represented as a linear function of \(\ln(r)\), \(\ln\left(\frac{\Delta V}{\Delta r}\right) = a \, \ln(r) + b\). What are the constants \( a \) and \( b \) in terms of \( C \) and \( n \)?\(^1\) b) Place the
Transcribed Image Text:**Problem 3:** It is postulated that the radial electric field of a group of charges falls off as \( E_r = \frac{C}{r^n} \), where \( C \) is a constant, \( r \) is the distance from the center of the group, and \( n \) is an unknown exponent. To test this hypothesis, you make a field probe consisting of two needle tips spaced \(\Delta r = 1.0 \, \text{mm}\) apart. You orient the needles so that a line between the tips points to the center of the charges, then use a voltmeter to read the potential difference between the tips. After you take measurements at several distances from the center of the group, your data are as given in the table. Use an appropriate graph of the data to determine the constants \( C \) and \( n \). | Distance, \( r \) (cm) | Potential difference, \( \Delta V \) (mV) | |------------------------|------------------------------------------| | 2.0 | 34.7 | | 4.0 | 6.6 | | 6.0 | 2.1 | | 8.0 | 1.2 | | 10.0 | 0.6 | a) In this experiment you are measuring the change of potential \( \Delta V \) over small displacement \( \Delta r \). In the limit \( \Delta r \to 0 \), \( \Delta V/\Delta r \) would give you the derivative of \( V \) with respect to \( r \), which is the radial component of the electric field, \( E_r \) (strictly speaking, with the minus sign). Take the natural logarithm of both sides of \( \frac{\Delta V}{\Delta r} = \frac{C}{r^n} \) to show that \(\ln\left(\frac{\Delta V}{\Delta r}\right)\) can be represented as a linear function of \(\ln(r)\), \(\ln\left(\frac{\Delta V}{\Delta r}\right) = a \, \ln(r) + b\). What are the constants \( a \) and \( b \) in terms of \( C \) and \( n \)?\(^1\) b) Place the
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