To express the SI unit of inductance, the henry (H), in terms of other physical quantities, we can derive its relationships step-by-step.
### Step-by-Step Solution:
1. **Understanding Inductance**:
- Inductance (L) is defined as the ratio of the magnetic flux (Φ) linked with a coil to the current (I) flowing through it.
- The formula is given by:
\[
L = \frac{\Phi}{I}
\]
2. **Unit of Magnetic Flux**:
- The SI unit of magnetic flux (Φ) is the weber (Wb).
- The SI unit of current (I) is the ampere (A).
- Therefore, substituting the units into the formula gives:
\[
[L] = \frac{\text{Wb}}{\text{A}} = \text{Wb/A}
\]
- This means that one henry can be expressed as:
\[
1 \, \text{H} = 1 \, \text{Wb/A}
\]
3. **Using Voltage and Time**:
- The relationship between voltage (V), inductance (L), and the rate of change of current (di/dt) is given by:
\[
V = L \frac{di}{dt}
\]
- Rearranging gives:
\[
L = \frac{V}{di/dt}
\]
- The unit of voltage (V) is joules per coulomb (J/C), and since 1 coulomb = 1 ampere × 1 second, we can express this as:
\[
L = \frac{J/C}{1/A} = \frac{J \cdot s}{A^2}
\]
- Thus, we can express henry in terms of joules and amperes:
\[
1 \, \text{H} = \frac{\text{J}}{\text{A}^2}
\]
4. **Using Energy and Resistance**:
- The energy (U) stored in an inductor is given by:
\[
U = \frac{1}{2} L I^2
\]
- Rearranging gives:
\[
L = \frac{2U}{I^2}
\]
- The unit of energy (U) is joules (J), so substituting gives:
\[
1 \, \text{H} = \frac{2 \, \text{J}}{\text{A}^2}
\]
5. **Using Resistance and Time**:
- The energy stored in an inductor can also be expressed in terms of resistance (R) and time (t):
\[
U = I^2 R t
\]
- Thus, we can relate L to R and t:
\[
L = R t
\]
- The unit of resistance (R) is ohms (Ω), and time (t) is seconds (s), leading to:
\[
1 \, \text{H} = \Omega \cdot s
\]
### Summary of Relationships:
- **Henry in terms of magnetic flux and current**:
\[
1 \, \text{H} = \frac{\text{Wb}}{\text{A}}
\]
- **Henry in terms of voltage and time**:
\[
1 \, \text{H} = \frac{\text{V} \cdot \text{s}}{\text{A}} = \frac{\text{J} \cdot \text{s}}{\text{A}^2}
\]
- **Henry in terms of energy and current**:
\[
1 \, \text{H} = \frac{\text{J}}{\text{A}^2}
\]
- **Henry in terms of resistance and time**:
\[
1 \, \text{H} = \Omega \cdot s
\]
