/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 A particle is placed in a field ... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 22

A particle is placed in a field characterized by a value of gravitational potential given by \(V=-k x y\), where \(k\) is a constant. If \(\vec{E}_{g}\) is the gravitational field then, (A) \(\vec{E}_{g}=k(x \hat{i}+y \hat{j})\) and is conservative in nature. (B) \(\vec{E}_{g}=k(y \hat{i}+x \hat{j})\) and is conservative in nature. (C) \(\vec{E}_{g}=k(x \hat{i}+y \hat{j})\) and is non-conservative in nature (D) \(\vec{E}_{g}=k(y \hat{i}+x \hat{j})\) and is non-conservative in nature.

Short Answer

Expert verified
The correct option is (B) \(\vec{E}_{g}=k(y \hat{i}+x \hat{j})\) and the field is conservative in nature.

Step by step solution

01

Calculate the Gravitational Field Vector

The gravitational field is the negative gradient of the gravitational potential. In our case, the potential is two-dimensional \(V=-k x y\), hence the gravitational field vector will be \(\vec{E}_{g}= -(dV/dx \hat{i} + dV/dy \hat{j})\). On differentiating the potential with respect to x and y, we obtain \(\vec{E}_{g}= -(-k y \hat{i} + -k x \hat{j}) = k(y \hat{i} + x \hat{j})\)
02

Check if the Field is Conservative

A field is conservative if its curl is zero. The curl of a two-dimensional field \(\vec{E}_{g}= P\hat{i} + Q\hat{j}\) is defined as \(curl \vec{E}_{g}= (dQ/dx - dP/dy)\hat{k}\). For our field, \(P= ky, Q= kx\). Hence, \(curl \vec{E}_{g}= (dkx/dx - dky/dy)\hat{k} = 0\hat{k}\), meaning that this field is conservative in nature.

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