SAN FRANCISCO – A federal magistrate has ordered Vice President Gore to give testimony in a trademark lawsuit against his new cable television venture, despite objections from the company’s lawyers that the request amounted to harassment.

Mr. Gore’s television service for young adults, Current TV, is being sued in an Ohio federal court by a company that provides broadband Internet service over power lines, Current Communications Group of Maryland.

In a three-page order issued Wednesday, the magistrate, Timothy Hogan, said he was “unimpressed” with the TV network’s assertion that Mr. Gore, who serves as Current TV’s chairman, was being harassed.

“The Defendant company is not a Fortune 500 company, but a small company, and the chairman here is also the founder. It is also significant that a public statement, the source of which is the chairman himself, indicates that he spends 75% of his time working on company business,” Magistrate Hogan, who sits in Cincinnati, wrote.

He also noted that the plaintiffs in the case suggested that Mr. Gore, who played a role in the early development of the Internet, “has the most intimate knowledge of the convergence between television and the Internet.”

Last year, the federal judge overseeing the case, Sandra Beckwith, rejected a request to block Mr. Gore’s network from using the “Current TV” name. She said there was little chance of confusion with the plaintiffs’ business, which focuses primarily on delivering broadband Web access to homes.

Judge Beckwith’s ruling gave no immediate relief to the Maryland firm, but the case was allowed to proceed.

In the new order, Magistrate Hogan rejected a suggestion from Current TV’s lawyers that all questions for Mr. Gore be asked and answered in writing, but agreed to limit the length of the former vice president’s deposition to six hours.

The magistrate said one reason for the deposition was to question Mr. Gore about e-mails he exchanged with the other founder of Current TV, Joel Hyatt, a legal services entrepreneur who once ran for the Senate from Ohio. He serves as the San Francisco based network’s CEO.

A spokesman for Current TV, Alex Dolan, had no comment yesterday on the legal developments. Earlier, he noted that more than 300 businesses in America have the word “Current” in their names.

The ongoing litigation is awkward for the Internet search firm Google. It has a stake reported at $100 million in the broadband-over-power-lines firm and a “content partnership” that helps produce segments on Internet search trends for Mr. Gore’s TV venture.

Magistrate Hogan’s order indicated that before launching his network, Mr. Gore discussed its naming with a cofounder of Google, Sergey Brin. He was also present at a press conference announcing the new channel.

Investors in the broadband firm, which include the investment bank Goldman Sachs and a venture associated with a cable TV company, Liberty Media, have ample resources for protracted litigation.

Current TV is facing a similar lawsuit from Minnesota Public Radio, which runs a radio station and Internet music feed called “The Current.”