The gravitational field in a region is given by \(\vec E\) \(=(5\,N / kg)\, \hat i + (12\,N / kg)\,\hat j\) If the potential at the origin is taken to be zero, then the ratio of the potential at the points \((12\,m, 0)\) and \((0, 5\,m)\) is

\({E_x} = 5\,N/kg\,and\,{E_y} = 12\,N/kg\)

Gravitational potential

                   \(=Gravitational\, field\times\,distance\)

\(\therefore {V_{\left( {12m,0} \right)}} = {E_x} \times 12\,J/kg\)

\(and\,{V_{\left( {0.5m} \right)}} = {E_y} \times 5\,\,J/kg\)

\(\left( {Give\,:\,potential\,at\,the\,origin\,is\,zero} \right)\)

\(\therefore \frac{{{V_{\left( {12\,m,0} \right)}}}}{{{V_{\left( {0,5m} \right)}}}} = \frac{{{E_x} \times 12}}{{{E_y} \times 5}} = \frac{{5 \times 12}}{{12 \times 5}} = 1\)