One end of a copper rod of uniform cross - section and of length 1.5 m is kept in contact with ice and the other end with water at 100^(@)C. At what point along its length should a temperature of 200^(@)C be maintained so that in steady state, the mass of ice melting be equal to that of the steam produced in the same interval of time ? Assume that the whole system is insulated from the surroundings.

One end of a copper rod of uniform cross - section and of length 1.5 m

1.	One end of a copper rod of uniform cross - section and of length 1.5 m is kept in contact with ice and the other end with water at 100^(@)C. At what point along its length should a temperature of 200^(@)C be maintained so that in steady state, the mass of ice melting be equal to that of the steam produced in the same interval of time ? Assume that the whole system is insulated from the surroundings.
Answer» Solution : If the POINT is at a distance x from water at `100^(@)C`, heat conducted to ice in time t, `Q_("ice")=KA((200-0))/((1.5-x))xxt` So ice melted by this heat `m_("ice")=Q_("ice")/L_(F)=(KA)/80((200-0))/((1.5-x))xxt` SIMILARLY heat conducted by the rod to the water at `100^(@)C` in time t, `Q_("water")=KA((200-100))/xt` SO steam formed by this heat `m_("steam")=Q_("water")/L_(v)=KA((200-100))/(540xxx)t` According to GIVEN problem, `m_("ice")=m_("steam"),i.e.` `200/(80(1.5-x))=100/(540xxx)rArrx=6/58m=10.34cm` i.e., `200^(@)C` temperature must be maintained at a distance 10.34 cm from water at `100^(@)C`