Show that the torque on an electric dipole placed in a uniform electric field is `tau=pxxE` independent of the origin about which torque is calculated.

Explanation

To show that the torque on an electric dipole placed in a uniform electric field is given by \( \tau = \mathbf{p} \times \mathbf{E} \) and is independent of the origin about which the torque is calculated, we can follow these steps:

### Step 1: Understand the Electric Dipole and Electric Field
An electric dipole consists of two equal and opposite charges, +Q and -Q, separated by a distance \( d \). The dipole moment \( \mathbf{p} \) is defined as:
\[ \mathbf{p} = Q \cdot \mathbf{d} \]
where \( \mathbf{d} \) is a vector pointing from the negative charge to the positive charge.

### Step 2: Analyze Forces on the Dipole
When the dipole is placed in a uniform electric field \( \mathbf{E} \), the positive charge experiences a force \( \mathbf{F}_+ = Q \mathbf{E} \) in the direction of the electric field, while the negative charge experiences a force \( \mathbf{F}_- = -Q \mathbf{E} \) in the opposite direction.

### Step 3: Calculate the Net Force
The net force \( \mathbf{F}_{\text{net}} \) acting on the dipole can be calculated as:
\[ \mathbf{F}_{\text{net}} = \mathbf{F}_+ + \mathbf{F}_- = Q \mathbf{E} - Q \mathbf{E} = 0 \]
Since the net force is zero, the dipole will not experience any translational motion, only rotational motion (torque).

### Step 4: Calculate the Torque on the Dipole
The torque \( \mathbf{\tau} \) on the dipole due to the forces can be calculated using the formula:
\[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \]
where \( \mathbf{r} \) is the position vector from the point about which we are calculating the torque to the point where the force is applied.

For the positive charge, the torque is:
\[ \mathbf{\tau}_+ = \mathbf{r}_+ \times \mathbf{F}_+ = \mathbf{r}_+ \times (Q \mathbf{E}) \]
For the negative charge, the torque is:
\[ \mathbf{\tau}_- = \mathbf{r}_- \times \mathbf{F}_- = \mathbf{r}_- \times (-Q \mathbf{E}) \]

### Step 5: Combine the Torques
The total torque \( \mathbf{\tau}_{\text{net}} \) is the sum of the torques due to both charges:
\[ \mathbf{\tau}_{\text{net}} = \mathbf{\tau}_+ + \mathbf{\tau}_- \]
This can be simplified to:
\[ \mathbf{\tau}_{\text{net}} = \left(\mathbf{r}_+ \times (Q \mathbf{E})\right) + \left(\mathbf{r}_- \times (-Q \mathbf{E})\right) \]

### Step 6: Express Torque in Terms of Dipole Moment
Using the definition of the dipole moment \( \mathbf{p} = Q \mathbf{d} \) and the geometry of the dipole, we can express the net torque as:
\[ \mathbf{\tau}_{\text{net}} = \mathbf{p} \times \mathbf{E} \]

### Step 7: Independence of the Origin
Since the net force on the dipole is zero, the torque calculated about any point (origin) will yield the same result. This is because the torque due to a net force of zero does not depend on the choice of origin. Thus, we conclude:
\[ \mathbf{\tau} = \mathbf{p} \times \mathbf{E} \]
is independent of the origin about which the torque is calculated.

### Final Result
The torque on an electric dipole in a uniform electric field is given by:
\[ \mathbf{\tau} = \mathbf{p} \times \mathbf{E} \]

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