Resistors and Conductivity - Revision history

Kjones313: /* Examples */

2015-12-01T00:02:37Z

Examples

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	<math>R_{equivalent} = R_1 + R_2 + R_3 + R_4</math>		<math>R_{equivalent} = R_1 + R_2 + R_3 + R_4</math>

	<math>R_{equivalent} = 65 \ Ω + 84 \ Ω + 73 \ Ω + 10 \ Ω</math>		<math>R_{equivalent} = 65 \ Ω + 84 \ Ω + 73 \ Ω + 10 \ Ω</math>

	<math>R_{equivalent} = 232 \ Ω</math>		<math>R_{equivalent} = 232 \ Ω</math>

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	<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{R_1}} + {\frac{1}{R_2}} + {\frac{1}{R_3}} + {\frac{1}{R_4}}</math>		<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{R_1}} + {\frac{1}{R_2}} + {\frac{1}{R_3}} + {\frac{1}{R_4}}</math>

	<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{10 \ Ω}} + {\frac{1}{20 \ Ω}} + {\frac{1}{30 \ Ω}} + {\frac{1}{40 \ Ω}}</math>		<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{10 \ Ω}} + {\frac{1}{20 \ Ω}} + {\frac{1}{30 \ Ω}} + {\frac{1}{40 \ Ω}}</math>

	<math>{\frac{1}{R_{equivalent}}} = {\frac{25}{120}} \ Ω</math>		<math>{\frac{1}{R_{equivalent}}} = {\frac{25}{120}} \ Ω</math>

	<math>R_{equivalent} = 4.8 \ Ω</math>		<math>R_{equivalent} = 4.8 \ Ω</math>

Kjones313: /* Examples */

2015-12-01T00:02:07Z

Examples

Kjones313: /* Resistors in Series */

2015-12-01T00:01:06Z

Resistors in Series

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	for ''n'' resistors in series.		for ''n'' resistors in series.


	Because R = L/(σA), if every resistor is composed of the same material and has the same cross-sectional area,		Because R = L/(σA), if every resistor is composed of the same material and has the same cross-sectional area,

Kjones313: /* Resistors in Parallel */

2015-12-01T00:00:45Z

Resistors in Parallel

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	for ''n'' resistors in parallel.		for ''n'' resistors in parallel.


	Because 1/R = (σA)/L, if every resistor is composed of the same material and has the same length,		Because 1/R = (σA)/L, if every resistor is composed of the same material and has the same length,

Kjones313: /* Resistors in Parallel */

2015-12-01T00:00:25Z

Resistors in Parallel

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	[[File:Resistors_in_Parallel.png\|300px\|center]]		[[File:Resistors_in_Parallel.png\|300px\|center]]

	Several resistors in parallel are, in practice, equivalent to a single resistor with a resistance that is the reciprocal of the sum of reciprocals of the individual resistances. In other words, ~~<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{R_1}} + {\frac{1}{R_2}} + {\frac{1}{R_3}} + ... + {\frac{1}{R_n}}</math> for ''n'' resistors in parallel.~~		Several resistors in parallel are, in practice, equivalent to a single resistor with a resistance that is the reciprocal of the sum of reciprocals of the individual resistances. In other words,

	Because 1/R = (σA)/L, if every resistor is composed of the same material and has the same length, <math>A_{equivalent} = A_1 + A_2 + A_3 + ... + A_n</math> for ''n'' resistors in parallel.		<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{R_1}} + {\frac{1}{R_2}} + {\frac{1}{R_3}} + ... + {\frac{1}{R_n}}</math>

			for ''n'' resistors in parallel.

			Because 1/R = (σA)/L, if every resistor is composed of the same material and has the same length,

			<math>A_{equivalent} = A_1 + A_2 + A_3 + ... + A_n</math>

			for ''n'' resistors in parallel.

	==Examples==		==Examples==

Kjones313: /* Resistors in Series */

2015-11-30T23:59:41Z

Resistors in Series

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	[[File:Resistors_in_Series.png\|300px\|center]]		[[File:Resistors_in_Series.png\|300px\|center]]

	Resistors in series are, in practice, equivalent to a single resistor with the combined resistance of its constituent resistors. In other words, ~~<math>R_{equivalent} = R_1 + R_2 + R_3 + ... + R_n</math> for ''n'' resistors in series.~~		Resistors in series are, in practice, equivalent to a single resistor with the combined resistance of its constituent resistors. In other words,

	Because R = L/(σA), if every resistor is composed of the same material and has the same cross-sectional area, <math>L_{equivalent} = L_1 + L_2 + L_3 + ... + L_n</math> for ''n'' resistors in series.		<math>R_{equivalent} = R_1 + R_2 + R_3 + ... + R_n</math>

			for ''n'' resistors in series.

			Because R = L/(σA), if every resistor is composed of the same material and has the same cross-sectional area,

			<math>L_{equivalent} = L_1 + L_2 + L_3 + ... + L_n</math>

			for ''n'' resistors in series.

	==Resistors in Parallel==		==Resistors in Parallel==

Kjones313: /* Relevant Equations */

2015-11-30T23:58:00Z

Relevant Equations

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	The conductivity of a material can be found using the equation		The conductivity of a material can be found using the equation

	<math>σ = \|q\|nu = {\frac{\vec{J}}{\vec{E}}}</math>		<math>σ = \|q\|nu = {\frac{\vec{J}}{\vec{E}}}</math>

	where \|''q''\| is the absolute value of the charge on each carrier, ''n'' is the number of charge carriers per m<sup>3</sup>, and ''u'' is the mobility of the charge carriers, '''J''' is the current density (I/A) in A/m<sup>2</sup>, and '''E''' is the electric field due to charges outside the material.		where \|''q''\| is the absolute value of the charge on each carrier, ''n'' is the number of charge carriers per m<sup>3</sup>, and ''u'' is the mobility of the charge carriers, '''J''' is the current density (I/A) in A/m<sup>2</sup>, and '''E''' is the electric field due to charges outside the material.


	Another equation used to quantify resistance relates it to certain properties of the material and geometric properties of the resistor itself:		Another equation used to quantify resistance relates it to certain properties of the material and geometric properties of the resistor itself:

	<math>R = {\frac{L}{σA}}</math>		<math>R = {\frac{L}{σA}}</math>

	where L is the length of the resistor, σ is the conductivity of the material, and A is the cross-sectional area of the resistor. This equation clearly demonstrates that resistivity and conductivity are inverses, as the conductivity constant can be found in the denominator.		where L is the length of the resistor, σ is the conductivity of the material, and A is the cross-sectional area of the resistor. This equation clearly demonstrates that resistivity and conductivity are inverses, as the conductivity constant can be found in the denominator.

Kjones313: /* Relevant Equations */

2015-11-30T23:57:33Z

Relevant Equations

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	The resistance of a material can be calculated in several ways. The most common method relates resistance to the potential difference and the conventional current of the circuit, using the equation		The resistance of a material can be calculated in several ways. The most common method relates resistance to the potential difference and the conventional current of the circuit, using the equation

	<math>R = {\frac{ΔV}{I}}</math>		<math>R = {\frac{ΔV}{I}}</math>

	where ΔV is the potential difference across the resistor and I is the conventional current running through the circuit.		where ΔV is the potential difference across the resistor and I is the conventional current running through the circuit.

Kjones313: /* Relevant Equations */

2015-11-30T23:57:13Z

Relevant Equations

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	==Relevant Equations==		==Relevant Equations==

	The resistance of a material can be calculated in several ways. The most common method relates resistance to the potential difference and the conventional current of the circuit, using the equation <math>R = {\frac{ΔV}{I}}</math> where ΔV is the potential difference across the resistor and I is the conventional current running through the circuit.		The resistance of a material can be calculated in several ways. The most common method relates resistance to the potential difference and the conventional current of the circuit, using the equation
			<math>R = {\frac{ΔV}{I}}</math>
			where ΔV is the potential difference across the resistor and I is the conventional current running through the circuit.


	The conductivity of a material can be found using the equation <math>σ = \|q\|nu = {\frac{\vec{J}}{\vec{E}}}</math> where \|''q''\| is the absolute value of the charge on each carrier, ''n'' is the number of charge carriers per m<sup>3</sup>, and ''u'' is the mobility of the charge carriers, '''J''' is the current density (I/A) in A/m<sup>2</sup>, and '''E''' is the electric field due to charges outside the material.		The conductivity of a material can be found using the equation
			<math>σ = \|q\|nu = {\frac{\vec{J}}{\vec{E}}}</math>
			where \|''q''\| is the absolute value of the charge on each carrier, ''n'' is the number of charge carriers per m<sup>3</sup>, and ''u'' is the mobility of the charge carriers, '''J''' is the current density (I/A) in A/m<sup>2</sup>, and '''E''' is the electric field due to charges outside the material.


	Another equation used to quantify resistance relates it to certain properties of the material and geometric properties of the resistor itself: <math>R = {\frac{L}{σA}}</math> where L is the length of the resistor, σ is the conductivity of the material, and A is the cross-sectional area of the resistor. This equation clearly demonstrates that resistivity and conductivity are inverses, as the conductivity constant can be found in the denominator.		Another equation used to quantify resistance relates it to certain properties of the material and geometric properties of the resistor itself:
			<math>R = {\frac{L}{σA}}</math>
			where L is the length of the resistor, σ is the conductivity of the material, and A is the cross-sectional area of the resistor. This equation clearly demonstrates that resistivity and conductivity are inverses, as the conductivity constant can be found in the denominator.

	==Ohmic vs. Non-Ohmic Resistors==		==Ohmic vs. Non-Ohmic Resistors==

Kjones313: /* References */

2015-11-27T02:15:01Z

References

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	#http://www.ceb.cam.ac.uk/data/images/groups/CREST/Teaching/impedence/series2.gif		#http://www.ceb.cam.ac.uk/data/images/groups/CREST/Teaching/impedence/series2.gif
	#https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Resistor_symbol_America.svg/2000px-Resistor_symbol_America.svg.png		#https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Resistor_symbol_America.svg/2000px-Resistor_symbol_America.svg.png


	Created by Kevin Jones, 26 November 2015		Created by Kevin Jones, 26 November 2015


	[[Category:Simple Circuits]]		[[Category:Simple Circuits]]

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	[[File:Resistors_in_Series_Example.gif\|350px\|left]] Because the circuit at left has no branches, the resistors can be considered in series. As such, the values of the resistance of the individual resistors can be summed to find the equivalent resistance. Thus,		[[File:Resistors_in_Series_Example.gif\|350px\|left]] Because the circuit at left has no branches, the resistors can be considered in series. As such, the values of the resistance of the individual resistors can be summed to find the equivalent resistance. Thus,

	<math>R_{equivalent} = R_1 + R_2 + R_3 + R_4</math>		<math>R_{equivalent} = R_1 + R_2 + R_3 + R_4</math>
	<math>R_{equivalent} = 65 \ Ω + 84 \ Ω + 73 \ Ω + 10 \ Ω</math>		<math>R_{equivalent} = 65 \ Ω + 84 \ Ω + 73 \ Ω + 10 \ Ω</math>
	<math>R_{equivalent} = 232 \ Ω</math>		<math>R_{equivalent} = 232 \ Ω</math>


	Using this equivalent resistance and the given voltage (25 V), it is possible to find the value of the current running through the circuit after passing through the resistors.		Using this equivalent resistance and the given voltage (25 V), it is possible to find the value of the current running through the circuit after passing through the resistors.

	<math>I = {\frac{ΔV}{R_{equivalent}}} = {\frac{25 \ V}{232 \ Ω}} \approx 0.108 \ A</math>		<math>I = {\frac{ΔV}{R_{equivalent}}} = {\frac{25 \ V}{232 \ Ω}} \approx 0.108 \ A</math>

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	[[File:Resistors_in_Parallel_Example.gif\|350 px\|right]] Because the circuit at right has several possible paths of current, the resistors can be considered in parallel.		[[File:Resistors_in_Parallel_Example.gif\|350 px\|right]] Because the circuit at right has several possible paths of current, the resistors can be considered in parallel.
	As a result, <math>{\frac{1}{R_{equivalent}}} = {\frac{1}{R_1}} + {\frac{1}{R_2}} + {\frac{1}{R_3}} + {\frac{1}{R_4}}</math>		As a result,

			<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{R_1}} + {\frac{1}{R_2}} + {\frac{1}{R_3}} + {\frac{1}{R_4}}</math>
	<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{10 \ Ω}} + {\frac{1}{20 \ Ω}} + {\frac{1}{30 \ Ω}} + {\frac{1}{40 \ Ω}}</math>		<math>{\frac{1}{R_{equivalent}}} = {\frac{1}{10 \ Ω}} + {\frac{1}{20 \ Ω}} + {\frac{1}{30 \ Ω}} + {\frac{1}{40 \ Ω}}</math>
	<math>{\frac{1}{R_{equivalent}}} = {\frac{25}{120}} \ Ω</math>		<math>{\frac{1}{R_{equivalent}}} = {\frac{25}{120}} \ Ω</math>
	<math>R_{equivalent} = 4.8 \ Ω</math>		<math>R_{equivalent} = 4.8 \ Ω</math>


	The voltage is again provided, and so it is possible to determine the total current I<sub>T</sub> using ΔV and R<sub>equivalent</sub>:		The voltage is again provided, and so it is possible to determine the total current I<sub>T</sub> using ΔV and R<sub>equivalent</sub>:

	<math>I_T = {\frac{ΔV}{R_{equivalent}}} = {\frac{24 \ V}{4.8 \ Ω}} = 5 \ A</math>		<math>I_T = {\frac{ΔV}{R_{equivalent}}} = {\frac{24 \ V}{4.8 \ Ω}} = 5 \ A</math>