Introduction:
In order to understand why the trans isomer has a greater dipole moment than the cis isomer, we need to first understand the concept of dipole moment and how it is influenced by the molecular structure.

Dipole Moment:
Dipole moment is a measure of the separation of positive and negative charges within a molecule. It is a vector quantity, meaning it has both magnitude and direction. The dipole moment is directly proportional to the charge and the distance between the charges.

Cis and Trans Isomers:
Cis and trans isomers are two types of stereoisomers that differ in the arrangement of substituent groups around a double bond. In the cis isomer, the substituent groups are on the same side of the double bond, whereas in the trans isomer, the substituent groups are on opposite sides of the double bond.

Opposite Inductive Character:
Inductive character refers to the ability of a substituent group to donate or withdraw electron density through sigma bonds. When two groups have opposite inductive character, it means that one group is electron-donating, while the other is electron-withdrawing.

Dipole Moment and Inductive Effect:
The dipole moment of a molecule is influenced by the polarity of the bonds within the molecule. The polarity of a bond is determined by the difference in electronegativity between the atoms involved in the bond. The greater the difference in electronegativity, the more polar the bond.

Explanation:
In the case of cis and trans isomers, the dipole moment is influenced by the opposite inductive character of the substituent groups. Let's consider a hypothetical example of a molecule with two substituent groups, A and B, attached to a double bond.

Cis Isomer:
In the cis isomer, both substituent groups (A and B) are on the same side of the double bond. If group A is an electron-donating group and group B is an electron-withdrawing group, the inductive effect of the two groups will partially cancel each other out. As a result, the overall dipole moment of the cis isomer will be smaller than if both groups had the same inductive character.

Trans Isomer:
In the trans isomer, the substituent groups (A and B) are on opposite sides of the double bond. This means that the inductive effect of the two groups will not cancel each other out, but rather add up. If group A is an electron-donating group and group B is an electron-withdrawing group, the overall dipole moment of the trans isomer will be greater than that of the cis isomer.

Conclusion:
In summary, the trans isomer has a greater dipole moment than the cis isomer when the two substituent groups have opposite inductive character. This is because the inductive effect of the substituent groups in the trans isomer adds up, resulting in a larger overall dipole moment.