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Physics-

General

Easy

Question

The variation of gravitational field E and potential $ϕ$ as a function of x from the center of a uniform spherical mass distribution of radius R is correctly represented, from the following plots, by

$E_{3} and ϕ_{4}$
$E_{2} and ϕ_{4}$
$E_{1} and ϕ_{1}$
$E_{1} and ϕ_{2}$

The correct answer is: $E subscript 2 end subscript text and end text ϕ subscript 4 end subscript$

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$A equals open square brackets table row cell I subscript 1 end subscript blank m subscript 1 end subscript blank n subscript 1 end subscript end cell row cell l subscript 2 end subscript blank m subscript 2 end subscript blank n subscript 2 end subscript end cell row cell l subscript 3 end subscript blank m subscript 3 end subscript blank n subscript 3 end subscript end cell end table close square brackets text and end text B equals open square brackets table row cell p subscript 1 end subscript end cell cell q subscript 1 end subscript end cell cell r subscript 1 end subscript end cell row cell p subscript 2 end subscript end cell cell q subscript 2 end subscript end cell cell r subscript 2 end subscript end cell row cell p subscript 3 end subscript end cell cell q subscript 3 end subscript end cell cell r subscript 3 end subscript end cell end table close square brackets$ where $p subscript i end subscript comma q subscript i end subscript comma r subscript i end subscript$ are the co-factors of the elements $I subscript i end subscript comma m subscript i end subscript comma n subscript i end subscript text end text$ for $i equals 1 , 2 comma 3. text If end text open parentheses I subscript 1 end subscript comma m subscript 1 end subscript comma n subscript 1 end subscript close parentheses comma open parentheses I subscript 2 end subscript comma m subscript 2 end subscript comma n subscript 2 end subscript close parentheses text and end text open parentheses l subscript 3 end subscript comma m subscript 3 end subscript comma n subscript 3 end subscript close parentheses$ are the direction cosines of of three mutually perpendicular lines then $open parentheses p subscript 1 end subscript comma q subscript 1 end subscript comma r subscript 1 end subscript close parentheses comma open parentheses p subscript 2 end subscript comma q subscript 2 end subscript comma r subscript 2 end subscript close parentheses text end text$ and $text end text open parentheses p subscript 3 end subscript comma q subscript 3 end subscript comma r subscript 3 end subscript close parentheses$ are

$A equals open square brackets table row cell I subscript 1 end subscript blank m subscript 1 end subscript blank n subscript 1 end subscript end cell row cell l subscript 2 end subscript blank m subscript 2 end subscript blank n subscript 2 end subscript end cell row cell l subscript 3 end subscript blank m subscript 3 end subscript blank n subscript 3 end subscript end cell end table close square brackets text and end text B equals open square brackets table row cell p subscript 1 end subscript end cell cell q subscript 1 end subscript end cell cell r subscript 1 end subscript end cell row cell p subscript 2 end subscript end cell cell q subscript 2 end subscript end cell cell r subscript 2 end subscript end cell row cell p subscript 3 end subscript end cell cell q subscript 3 end subscript end cell cell r subscript 3 end subscript end cell end table close square brackets$ where $p subscript i end subscript comma q subscript i end subscript comma r subscript i end subscript$ are the co-factors of the elements $I subscript i end subscript comma m subscript i end subscript comma n subscript i end subscript text end text$ for $i equals 1 , 2 comma 3. text If end text open parentheses I subscript 1 end subscript comma m subscript 1 end subscript comma n subscript 1 end subscript close parentheses comma open parentheses I subscript 2 end subscript comma m subscript 2 end subscript comma n subscript 2 end subscript close parentheses text and end text open parentheses l subscript 3 end subscript comma m subscript 3 end subscript comma n subscript 3 end subscript close parentheses$ are the direction cosines of of three mutually perpendicular lines then $open parentheses p subscript 1 end subscript comma q subscript 1 end subscript comma r subscript 1 end subscript close parentheses comma open parentheses p subscript 2 end subscript comma q subscript 2 end subscript comma r subscript 2 end subscript close parentheses text end text$ and $text end text open parentheses p subscript 3 end subscript comma q subscript 3 end subscript comma r subscript 3 end subscript close parentheses$ are

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