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Exercise:
A fixed resistor R and a variable resistor R_v are connected in series to a voltage supply Delta V. abcliste abc Derive a formal expression for the power dissipated in the variable resistor. abc Show that the power dissipated in the variable resistor is greatest for R_v R. abcliste

Solution:
abcliste abc The power dissipated in the varialbe resistor can be expressed as P_v R_v I^ R_vleftfracDelta VR+R_vright^ abc In order to find the maximum we can derive P_v with respect to R_v using the product rule: fracddP_vddR_v leftfracDelta VR+R_vright^ - R_vfracleftDelta Vright^leftR+R_vright^ At the maximum this expression has to be equal to zero: fracleftDelta Vright^leftR+R_vright^ - R_vfracleftDelta Vright^leftR+R_vright^ Rearranging the terms multiplying by R+R_v^ and dividing by Delta V^ leads to R_v R+R_v The solution R_vR easily follows from this. abcliste
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Exercise:
A fixed resistor R and a variable resistor R_v are connected in series to a voltage supply Delta V. abcliste abc Derive a formal expression for the power dissipated in the variable resistor. abc Show that the power dissipated in the variable resistor is greatest for R_v R. abcliste

Solution:
abcliste abc The power dissipated in the varialbe resistor can be expressed as P_v R_v I^ R_vleftfracDelta VR+R_vright^ abc In order to find the maximum we can derive P_v with respect to R_v using the product rule: fracddP_vddR_v leftfracDelta VR+R_vright^ - R_vfracleftDelta Vright^leftR+R_vright^ At the maximum this expression has to be equal to zero: fracleftDelta Vright^leftR+R_vright^ - R_vfracleftDelta Vright^leftR+R_vright^ Rearranging the terms multiplying by R+R_v^ and dividing by Delta V^ leads to R_v R+R_v The solution R_vR easily follows from this. abcliste
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Attributes & Decorations
Branches
Direct Current
Tags
internal resistance, resistor, resistor circuit, series resistors
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Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Algebraic
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Decoration
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